Unit 16 · Geometry

A rectangle is easy: count the squares inside. A rectangle 8 cm wide and 5 cm tall covers 8 × 5 = 40 square centimeters. A parallelogram is a four-sided shape whose opposite sides are parallel, like a rectangle that has been pushed sideways. It has no square corners, so counting squares gets messy at the slanted ends.

Here is the trick. Draw a straight line from one top corner down to the bottom edge, at a right angle. That cuts off a triangle. Slide the triangle across to the other end of the shape. The two pieces fit together perfectly into a rectangle. Moving a piece never changes how much space it covers, so the parallelogram and the rectangle have the same area.

The rectangle’s width is the parallelogram’s base. Its height is the parallelogram’s height, which is the straight-up distance from the base to the opposite side, not the slanted side. So area = base × height. A parallelogram with base 8 cm and height 5 cm has area 8 × 5 = 40 cm², exactly like the rectangle.

The most common mistake is to multiply the base by the slanted side instead of the height. If that parallelogram’s slanted side is 6 cm, some students write 8 × 6 = 48 cm². That is wrong. The slanted side is longer than the height, because it leans. Always look for the measurement that makes a right angle with the base.

Take any triangle. Make an exact copy, turn the copy upside down and fit the two together along one side. The two triangles form a parallelogram with the same base and the same height as the triangle. Since the parallelogram’s area is base × height, and the triangle is half of it, the triangle has area 1/2 × base × height.

Try it with a triangle whose base is 10 cm and whose height is 6 cm. Two copies make a parallelogram with area 10 × 6 = 60 cm². So the triangle covers half of that: 1/2 × 60 = 30 cm². You can multiply in any order. Some students find it easier to halve first: half of 10 is 5, then 5 × 6 = 30.

As with parallelograms, the height must meet the base at a right angle. In a right triangle, the two sides that form the square corner are the base and the height, so the job is easy. A right triangle with legs 6 cm and 8 cm has area 1/2 × 6 × 8 = 24 cm². The long slanted side, 10 cm, is not used at all.

Two mistakes show up again and again. One is forgetting the half, which gives an answer twice too big. The other is using a slanted side as the height. In a tall, leaning triangle the height may even fall outside the shape. Draw a dotted line straight down from the top corner to the line of the base and measure that.

A trapezoid is a four-sided shape with exactly one pair of parallel sides. Those two parallel sides are its bases, and they are usually different lengths. Imagine a trapezoid with a short top base of 6 cm, a long bottom base of 10 cm and a height of 4 cm. It is not a rectangle and not a triangle, but it can be cut into both.

One way: draw a vertical line down from the top corner to the bottom. That gives a rectangle 6 cm by 4 cm and a triangle with base 4 cm (the extra 10 − 6) and height 4 cm. Rectangle: 6 × 4 = 24. Triangle: 1/2 × 4 × 4 = 8. Total: 24 + 8 = 32 cm².

Another way: make a copy, flip it upside down and fit it next to the original. The two trapezoids form a parallelogram with base 6 + 10 = 16 cm and height 4 cm, area 16 × 4 = 64 cm². The trapezoid is half of that: 32 cm². Same answer. This second way gives the formula: area = 1/2 × (base 1 + base 2) × height. You can read it as the average of the two bases, times the height.

Check it on the example: the average of 6 and 10 is 8, and 8 × 4 = 32. The common mistake is multiplying the two bases together, 6 × 10, which makes no sense; you add them. Another is forgetting to halve the sum, which doubles the answer.

The distance across a circle through its center is the diameter. Half of that, from the center to the edge, is the radius. The distance all the way around is the circumference. Maya’s shoelace showed that circumference ÷ diameter comes out the same for every circle. That number is π, about 3.14. So circumference = π × diameter, or C = πd.

A dinner plate has diameter 10 inches. Its circumference is about 3.14 × 10 = 31.4 inches. A bicycle wheel with diameter 26 inches rolls forward one circumference each turn: 3.14 × 26 = 81.64, or about 82 inches per turn. If you know the radius instead, double it first. A radius of 7 cm means a diameter of 14 cm and a circumference of about 3.14 × 14 = 43.96 cm.

You can also go backward. A round table has a circumference of 47.1 inches. Its diameter is 47.1 ÷ 3.14 = 15 inches. Check: 3.14 × 15 = 47.1. The mistake to watch for is mixing up radius and diameter. If a problem gives the radius and you multiply it by π, your answer is half of the real circumference.

Cut a paper circle into 16 thin wedges, like pizza slices. Lay them in a row, point up, point down, point up, point down. They fit together into a bumpy shape that is almost a parallelogram. The more slices you cut, the flatter the bumps get. With a thousand slices it is a rectangle for all practical purposes.

What are that rectangle’s measurements? Its height is the length of one slice from point to crust, which is the radius r. Its width is half the crust, because half the wedges point up and half point down. Half the circumference is 1/2 × 2πr = πr. So the area is width × height = πr × r = π × r². That is the formula: area = πr².

A circle with radius 5 cm has area 3.14 × 5 × 5 = 3.14 × 25 = 78.5 cm². If a problem gives the diameter, halve it first. A pizza 12 inches across has a radius of 6 inches and an area of 3.14 × 36 = 113.04 square inches. The number gets squared, not π, and not the diameter.

This formula settles arguments. Is one 10-inch pizza more food than two 6-inch pizzas? The 10-inch has radius 5, area 3.14 × 25 = 78.5 in². Each 6-inch has radius 3, area 3.14 × 9 = 28.26 in², so two of them cover 56.52 in². The single 10-inch pizza is bigger. Doubling a radius does not double the area; it multiplies it by four.

Take a cereal box, cut along some of its edges, and press it flat. You get six rectangles joined together. That flat pattern is a net. Fold it back up and it becomes the box again. Every solid with flat sides has a net, and a net shows every face of the solid at once, laid out where you can measure it.

A box 4 cm long, 3 cm wide and 2 cm tall has three pairs of matching faces: two that are 4 by 3, two that are 4 by 2, and two that are 3 by 2. The front matches the back, the top matches the bottom, and the left matches the right. A cube’s net is six equal squares. A triangular prism unfolds into two triangles and three rectangles.

Pyramids are different. A square pyramid has one square base and four triangles that meet at the top point, so its net is a square with a triangle on each side. Count faces before you compute anything: a net with the wrong number of faces gives the wrong surface area every time.

One warning: not every arrangement of six squares folds into a cube. If four squares sit in a row and two more hang off the same square on the same side, two faces will overlap when you fold. Try it with paper. Testing a net by folding is the fastest way to catch a mistake.

Surface area is how much material it takes to cover the outside of a solid: wrapping paper for a gift, cardboard for a box, paint for a wall. To find it, find the area of every face and add. The net makes sure you do not skip one. Surface area is measured in square units, because it is an area.

Take a box 5 in long, 4 in wide and 3 in tall. Its faces come in pairs. Top and bottom: 5 × 4 = 20 each, so 40. Front and back: 5 × 3 = 15 each, so 30. Left and right: 4 × 3 = 12 each, so 24. Total surface area: 40 + 30 + 24 = 94 in². A cube with 3 cm edges is easier: six faces of 3 × 3 = 9, so 6 × 9 = 54 cm².

A triangular prism takes one extra step. Say its triangle ends have base 6 cm and height 4 cm, with sides 5 cm, 5 cm and 6 cm, and the prism is 10 cm long. Each triangle: 1/2 × 6 × 4 = 12, so the two ends make 24. The three rectangles are each 10 cm long and as wide as one triangle side: 5 × 10 + 5 × 10 + 6 × 10 = 50 + 50 + 60 = 160. Total: 24 + 160 = 184 cm².

Common mistakes: counting only one of each pair of faces, which gives half the answer; forgetting the two end faces of a prism; and confusing surface area with volume. If your answer is in cubic units, you have measured the wrong thing.

A square pyramid has a square base and four triangular faces that meet at a point. To find its surface area, add the base to the four triangles. The triangles have a base equal to the side of the square. Their height is measured along the face, from the middle of a base edge up to the top point. That measurement is the slant height.

Take a pyramid with a base 6 cm on a side and a slant height of 5 cm. Base: 6 × 6 = 36 cm². Each triangle: 1/2 × 6 × 5 = 15 cm². Four of them: 4 × 15 = 60. Surface area: 36 + 60 = 96 cm². If the pyramid is a tent and you only need fabric for the sides, leave out the base: 60 cm².

Slant height is not the same as the pyramid’s height. The height goes straight up from the center of the base to the point, inside the solid. The slant height leans along the outside. The slant height is always the longer of the two. Using the height in the triangle formula gives an answer that is too small.

Bigger example: base 10 m, slant height 8 m. Base 10 × 10 = 100. Triangles 4 × (1/2 × 10 × 8) = 4 × 40 = 160. Total 260 m². Forgetting to halve the triangles gives 4 × 80 + 100 = 420, which is wrong. Forgetting the base gives 160, which is only right if the problem asks for the sides alone.

Volume is the amount of space inside a solid, measured in cubic units. A cubic centimeter is a cube 1 cm on each edge. A box 5 cm by 4 cm by 3 cm holds a bottom layer of 5 × 4 = 20 cubes, and there are 3 layers, so 20 × 3 = 60 cubic centimeters, written 60 cm³.

That idea works for every prism and every cylinder: volume = area of the base × height. The base is one flat end, and the height is how tall the stack is. For the box, the base area is 20 and the height is 3. For a triangular prism whose triangle end has base 6 cm and height 4 cm, the base area is 1/2 × 6 × 4 = 12 cm². If the prism is 10 cm long, the volume is 12 × 10 = 120 cm³.

A cylinder is a stack of circles. Its base area is πr². A can with radius 3 cm and height 10 cm has volume 3.14 × 3² × 10 = 3.14 × 9 × 10 = 282.6 cm³. Smaller can: radius 2 cm, height 7 cm gives 3.14 × 4 × 7 = 87.92 cm³.

Watch two things. First, square the radius, not the diameter. Using a diameter of 4 for the small can gives 3.14 × 16 × 7 = 351.68, four times too big. Second, keep surface area and volume separate. The 5 by 4 by 3 box has surface area 94 cm² and volume 60 cm³; they answer different questions.

Take a cone and a cylinder with the same circular base and the same height. Fill the cone with water and pour it into the cylinder. It takes exactly three cones to fill the cylinder. So a cone’s volume is one third of the cylinder’s: V = 1/3 × π × r² × h.

A paper cone cup has radius 3 cm and height 4 cm. The cylinder around it would hold 3.14 × 9 × 4 = 113.04 cm³. The cone holds a third of that: 113.04 ÷ 3 = 37.68 cm³. You can also take the third early: 1/3 × 3.14 × 9 × 4 = 3.14 × 12 = 37.68. Another cone, radius 2 cm and height 6 cm: 1/3 × 3.14 × 4 × 6 = 3.14 × 8 = 25.12 cm³.

A sphere is a ball. Fit a sphere snugly inside a cylinder, so the cylinder is as tall as the sphere and just as wide. The sphere fills exactly two thirds of the cylinder. That cylinder has radius r and height 2r, so its volume is πr² × 2r = 2πr³, and two thirds of that is 4/3 πr³. A ball with radius 3 cm has volume 4/3 × 3.14 × 27 = 4 × 3.14 × 9 = 113.04 cm³.

Two errors are common. Forgetting the 1/3 on a cone gives the cylinder’s volume instead, three times too big. On a sphere, writing r² instead of r³ gives 4/3 × 3.14 × 9 = 37.68 for the 3 cm ball, which is far too small. The sphere formula cubes the radius because volume has three dimensions.

Real problems rarely say ’find the surface area.’ They ask how much wrapping paper, how much paint, how much water. Ask yourself: am I covering the outside or filling the inside? Covering means surface area, in square units. Filling means volume, in cubic units. Wrapping paper covers, so it is surface area. Water fills, so it is volume.

A rectangular swimming pool is 20 m long, 8 m wide and 1.5 m deep. How much water fills it? That is volume: 20 × 8 × 1.5 = 240 cubic meters. If the pool’s floor and four walls need to be painted, that is surface area, but only five faces, since a pool has no top. Floor: 20 × 8 = 160. Long walls: 2 × (20 × 1.5) = 60. Short walls: 2 × (8 × 1.5) = 24. Total 244 m².

A soup can has radius 4 cm and height 10 cm. The paper label wraps around the curved side only. Unroll the label and it is a rectangle as tall as the can and as long as the circumference. Circumference: 2 × 3.14 × 4 = 25.12 cm. Label area: 25.12 × 10 = 251.2 cm². The soup inside is volume: 3.14 × 16 × 10 = 502.4 cm³.

Sometimes only part of the surface counts. A room gets paint on the walls but not the floor, a box of cereal has no lid at the factory before it is sealed, and a fish tank has no top. Sketch the net and cross out any face the problem leaves out before you add.

An angle measures how far one line turns from another, in degrees. A square corner is 90 degrees. A straight line is 180 degrees, because it is two square corners side by side. Those two numbers give us two important kinds of angle pairs.

Two angles whose measures add to 90 degrees are complementary. If a corner of a square is split by a diagonal line into a 35-degree angle and a second angle, the second one must be 90 − 35 = 55 degrees. Check: 35 + 55 = 90. The two angles do not have to touch. A 30-degree angle on one page and a 60-degree angle on another page are still complementary.

Two angles whose measures add to 180 degrees are supplementary. Picture a straight road with a side street branching off. The two angles the side street makes with the road add to 180. If one is 112 degrees, the other is 180 − 112 = 68 degrees. Check by adding: 112 + 68 = 180.

Students mix up the two words. One memory trick: C comes before S in the alphabet, and 90 comes before 180. Another common slip is subtracting from the wrong number, such as finding 90 − 112 and getting a negative angle. If your answer is negative or bigger than the total, you used the wrong pair.

Two straight lines cross like an X and make four angles. The angles across from each other, tip to tip, are called vertical angles. Vertical angles are always equal. The angles that share a side and sit next to each other are adjacent angles. Adjacent angles on a straight line are supplementary, because together they make a straight line of 180 degrees.

Suppose one angle of the X measures 70 degrees. The angle right next to it, along the same straight line, is 180 − 70 = 110 degrees. The angle across from the 70 is also 70. The angle across from the 110 is also 110. So the four angles are 70, 110, 70, 110, and all four add to 360, a full turn.

Why are vertical angles equal? Call the angles A, B, C and D going around the X. A and B are on a line, so A + B = 180. B and C are on a line too, so B + C = 180. Both A and C equal 180 − B, so A = C. No measuring needed; the straight lines force it.

A common mistake is to assume that adjacent angles are equal because they look similar. They are equal only if each is exactly 90 degrees, which happens when the lines are perpendicular. Otherwise one is smaller and one is larger, and only the pair across the X matches.

Many angle problems give a relationship instead of a number. Two angles are supplementary and one is twice the other. What are they? Call the smaller one x. The larger is 2x. Together they make 180, so x + 2x = 180, which is 3x = 180, so x = 60. The angles are 60 and 120. Check: 60 + 120 = 180, and 120 is twice 60.

The same plan works for complementary angles. One angle is four times its complement. Call the small one x, the large one 4x, and write x + 4x = 90. That is 5x = 90, so x = 18. The angles are 18 and 72. Check: 18 + 72 = 90, and 4 × 18 = 72.

Sometimes three or more angles share a line or a point. Three angles sit side by side along a straight line and measure 45, 65 and x. Then 45 + 65 + x = 180, so 110 + x = 180 and x = 70. If angles go all the way around a point, they add to 360 instead.

The main error is stopping too soon. Solving for x gives the small angle. If the question asks for the large angle, multiply. If it asks for both, list both. Always reread the question after you finish the algebra, and always add your answers back together to check the total.

Two lines are parallel if they run in the same direction and never meet, like railroad tracks. A third line that cuts across both is a transversal. Where it crosses each track it makes four angles, so eight angles in all. Because the two tracks point the same way, the eight angles come in only two sizes: a small one and a large one that add to 180.

Some pairs have names. Corresponding angles sit in the same position at each crossing, like the upper-left angle at the first track and the upper-left angle at the second. They are equal. Alternate interior angles lie between the two tracks on opposite sides of the transversal, forming a Z shape. They are equal too. Same-side interior angles lie between the tracks on the same side and add to 180.

Suppose a transversal makes a 130-degree angle at the top track. The corresponding angle at the bottom track is also 130. The angle next to it on the line is 180 − 130 = 50. Every angle in the picture is either 130 or 50. If one alternate interior angle is 55, its partner is 55, and the same-side interior angle across the transversal is 180 − 55 = 125.

This is the fact Eratosthenes used. The sun’s rays are parallel lines. The stick in Alexandria and the line from the Earth’s center to Syene form a transversal. The shadow angle at the stick equals the angle at the Earth’s center, because they are alternate interior angles. The rule works only when the lines really are parallel. If they lean even slightly, the equal pairs stop being equal.

Tear the three corners off a paper triangle and fit them together point to point. They always make a straight line. That is the famous rule: the three angles of any triangle add to 180 degrees. If two angles of a triangle are 48 and 67, the third is 180 − 48 − 67 = 65. Check: 48 + 67 + 65 = 180.

Here is why it is always true. Draw a triangle with corners A at the top, and B and C at the bottom. Through A, draw a line parallel to the bottom side BC. Now the side AB is a transversal crossing two parallel lines, so the angle at B has an equal alternate interior angle up at A, to the left. In the same way, the angle at C has an equal partner at A, to the right. The three angles at A, the left partner, the triangle’s own angle, and the right partner, sit along a straight line. Together they make 180. So angle B plus angle A plus angle C is 180.

A right triangle already spends 90 degrees on its square corner, so the other two angles add to 90 and are complementary. If one is 34 degrees, the other is 90 − 34 = 56. In an equilateral triangle, all three angles are equal, so each is 180 ÷ 3 = 60 degrees.

The common error is subtracting only one angle. Given 48 and 67, some students write 180 − 48 = 132 and stop. Subtract both, or add the two known angles first and subtract the sum. Another slip is forgetting that a right angle counts as one of the three.

Can you make a triangle from sticks of length 3, 4 and 8? Lay the 8 down. Now try to connect the 3 and the 4 above it so their ends meet. Even lying flat along the 8, they only reach 3 + 4 = 7. They never touch. No triangle. The two shorter sides must add to more than the longest side. That rule is the triangle inequality.

The rule says: the sum of any two sides is greater than the third side. It is enough to check the two shortest sides against the longest. Sides 5, 7 and 9: 5 + 7 = 12, which is more than 9, so yes. Sides 2, 3 and 6: 2 + 3 = 5, less than 6, so no. Sides 6, 6 and 12: 6 + 6 = 12, exactly equal, so no. The two short sides lie flat on the long one and the triangle collapses into a line.

The rule also tells you how long a missing side can be. Two sides are 5 and 7. The third side must be less than 5 + 7 = 12, or the other two cannot reach across it. It must also be more than 7 − 5 = 2, or the 5 and the third side together cannot reach across the 7. So the third side is any length strictly between 2 and 12: 3 works, 10 works, 2 and 12 do not.

This is why the shortest path between two points is a straight line. Any detour through a third point turns the trip into two sides of a triangle, and two sides always add to more than the third. Walking the diagonal across a park is shorter than walking around two edges for exactly this reason.

Give a builder three side lengths, say 5, 6 and 7 cm. Draw the 7 as the base. Set a compass to 5 and draw an arc from the left end, then set it to 6 and draw an arc from the right end. The arcs cross at exactly one point above the base. Connect it, and you have the triangle. Anyone who follows these steps gets the same triangle, just possibly flipped or turned. Three sides fix a triangle completely.

Now give a builder three angles instead, 30, 60 and 90 degrees. You can draw a tiny triangle with those angles or a huge one. All of them have the same shape, but they are different sizes. Three angles alone build many triangles, not one. You would need at least one side length to pin down the size.

Some instructions build nothing. Angles of 100, 50 and 40 degrees add to 190, more than 180, so no triangle has them. Sides 3, 4 and 8 fail the triangle inequality. And some sets fall in between. Two sides and the angle between them fix one triangle. Two sides and an angle not between them can sometimes fit two different triangles.

Before you start drawing, sort the request. Do the angles add to 180? Do the sides pass the triangle inequality? Is there at least one side length? Those three questions tell you whether to expect one triangle, many or none.

Cut a block of cheese and look at the cut face. That flat shape is a cross section. Slice a rectangular block straight across, parallel to one end, and the cross section is a rectangle the same size as that end. Slice a cylinder of cheese parallel to its round base and you get a circle. Slice the same cylinder straight down through its center, along its length, and you get a rectangle.

Pyramids and cones change with height. A square pyramid sliced parallel to its base gives a square, and the square gets smaller as the slice moves up toward the point. Sliced straight down through the top point and the middle of the base, the same pyramid gives a triangle. A cone sliced parallel to its base gives a circle; sliced through its tip and straight down, a triangle.

A cube sliced parallel to a face gives a square. But tilt the blade. Slicing a cube corner to corner through opposite edges gives a rectangle that is longer than it is wide. Slicing off one corner gives a triangle. Some slices of a cube even give a hexagon, a six-sided shape, when the blade passes through the middle of six edges.

Cross sections show up everywhere. A doctor’s scan is a stack of cross sections of a body. An architect’s floor plan is a horizontal cross section of a building. A slice of an orange is a circle because an orange is a sphere, and every straight slice of a sphere is a circle.

A scale drawing shows something real, like a room or a city, shrunk or enlarged so every length changes by the same amount. The scale tells you the match. A floor plan with a scale of 1 cm : 4 m means every centimeter on paper stands for 4 meters in the building. The shape stays the same; only the size changes.

To go from drawing to real, multiply. A wall that is 6 cm long on the plan is 6 × 4 = 24 m in real life. A wall that is 7.5 cm on the plan is 7.5 × 4 = 30 m. To go from real to drawing, divide. A real hallway of 50 m appears as 50 ÷ 4 = 12.5 cm on the plan. Check by going back: 12.5 × 4 = 50.

Maps use scales too. A road map with a scale of 1 inch : 20 miles shows two towns 180 miles apart as 180 ÷ 20 = 9 inches apart on the paper. A model car built at 1 : 24 is one twenty-fourth of the real size, so a real car 4.8 m long becomes a model 4.8 ÷ 24 = 0.2 m, or 20 cm.

The common error is going the wrong direction, dividing when you should multiply. Ask which is bigger. If the drawing is smaller than the real thing, real lengths must come out larger than drawing lengths. If your real wall comes out shorter than the drawing, you divided by mistake.

A scale factor is the single number every length gets multiplied by. Enlarging a photo by a scale factor of 3 makes every length three times as long. A rectangle 2 cm by 5 cm becomes 6 cm by 15 cm. But look at the areas. The original covers 2 × 5 = 10 cm². The copy covers 6 × 15 = 90 cm². The area is not 3 times bigger. It is 9 times bigger.

That always happens, because area has two dimensions and both get multiplied. Scale factor 3 makes the area 3 × 3 = 9 times bigger. Scale factor 2 makes it 2 × 2 = 4 times bigger. A garden of 12 m² enlarged by a scale factor of 2 becomes 12 × 4 = 48 m². Going the other way, a scale factor of 1/2 makes each length half as long and the area 1/2 × 1/2 = 1/4 as large.

This explains the pizza puzzle from the last chapter. A 12-inch pizza has twice the diameter of a 6-inch pizza, a scale factor of 2, so it has four times the area. It also matters for cost. If a poster twice as wide and twice as tall costs the same per square foot, it costs four times as much, not twice as much.

The mistake is to multiply the area by the scale factor itself. If a drawing at 1 cm : 4 m shows a room as 5 cm by 6 cm, the plan area is 30 cm². The real room is 20 m by 24 m, so 480 m². That is 30 × 16, because 4 × 4 = 16, not 30 × 4 = 120.

Slide a book across a table without turning it. Every corner of the book moves the same distance in the same direction. That move is a translation. The moved shape is called the image, and it is the same size and shape as the original, just somewhere else. On a coordinate grid, a translation is a rule like: add 3 to every x and subtract 2 from every y.

Take a triangle with corners at A(0, 0), B(3, 0) and C(0, 4). Translate it 2 units right and 1 unit up. Add 2 to each x and 1 to each y. A goes to A’(2, 1), B goes to B’(5, 1), and C goes to C’(2, 5). The little mark after each letter is read ’A prime’ and means the image of A. The new triangle has the same side lengths and the same angles as the old one.

Directions turn into signs. Right means add to x, left means subtract from x. Up means add to y, down means subtract from y. The point (−2, 5) moved 4 right and 3 down becomes (−2 + 4, 5 − 3) = (2, 2). Moving left 4 and up 3 instead would give (−6, 8).

The most common mistake is mixing the signs, adding when you should subtract. Another is applying the rule to only one corner. Every point must move by the same amounts, or the shape stretches and it is no longer a translation. Check by comparing one side before and after: it should be the same length and point the same way.

Hold a letter up to a mirror. The mirror image is the same size, but left and right are swapped. That is a reflection, and the mirror is the line of reflection. Every point of the image is exactly as far from the line as the original point, but on the opposite side. If a point sits on the line itself, it does not move at all.

On the coordinate grid, the two easiest mirrors are the axes. Reflecting across the x-axis flips the shape up to down: the x stays and the y changes sign. The point (3, −2) becomes (3, 2). Reflecting across the y-axis flips left to right: the y stays and the x changes sign. The point (3, −2) becomes (−3, −2).

Try a whole triangle: corners at (1, 1), (4, 1) and (1, 3). Reflect across the y-axis: (−1, 1), (−4, 1) and (−1, 3). The image is the same size, and every side is the same length. But a corner that used to be on the left is now on the right. If you read the corners going around, the direction reverses, like a clock running backward.

Watch which coordinate changes. Students often flip the wrong one. Think about what the mirror does. The x-axis is a horizontal line, so a flip across it moves points up or down, which is a change in y. The y-axis is vertical, so a flip across it moves points left or right, which is a change in x.

Pin one corner of a paper triangle to a board and spin the paper. That is a rotation. The pin is the center of rotation, and the amount of spin is the angle, such as 90 or 180 degrees. Counterclockwise is the usual direction unless a problem says clockwise. Every point stays exactly as far from the center as it was; it just swings to a new spot.

On the grid, rotating about the origin has simple rules. A 90-degree counterclockwise turn sends (x, y) to (−y, x). So (4, 1) goes to (−1, 4). A 180-degree turn sends (x, y) to (−x, −y), so (4, 1) goes to (−4, −1). A 90-degree clockwise turn sends (x, y) to (y, −x), so (4, 1) goes to (1, −4). The distance from the origin never changes: the point (4, 1) and all three images are the same distance from (0, 0).

You can check a 90-degree turn with a picture. From the origin, go 4 right and 1 up to reach (4, 1). Turn that path a quarter turn counterclockwise: right becomes up and up becomes left. Now go 4 up and 1 left, which is (−1, 4). It matches the rule.

The rule for 90 degrees is the one students mix up most. The coordinates swap and one of them changes sign. Which one? For counterclockwise, the new x is the negative of the old y. Test it with a point you can picture: (1, 0) on the right side of the grid should turn to (0, 1) at the top, and the rule gives (−0, 1) = (0, 1).

A projector throws a small image onto a big screen. Every length gets multiplied by the same number, and every angle stays exactly the same. That is a dilation. It has a center, the point the shape grows out from, and a scale factor. A scale factor bigger than 1 enlarges; a scale factor between 0 and 1 shrinks.

On the grid with the center at the origin, the rule is easy: multiply both coordinates by the scale factor. With scale factor 2, the point (2, 3) becomes (4, 6). With scale factor 3, the point (3, −1) becomes (9, −3). With scale factor 1/2, the point (8, 6) becomes (4, 3). A triangle with corners (0, 0), (2, 0) and (0, 1) dilated by 3 becomes (0, 0), (6, 0) and (0, 3): three times as wide and three times as tall.

Unlike slides, flips and turns, a dilation changes lengths. Every side of the image is the scale factor times the matching side of the original. But it does not change angles, and it does not change the shape. A dilated square is still a square. A dilated triangle has the same three angles as before.

Two mistakes to avoid. One is adding the scale factor instead of multiplying: (3, −1) with scale factor 3 is (9, −3), not (6, 2). The other is scaling only one coordinate, which stretches the shape in one direction and ruins its angles. Both coordinates get multiplied, every time.

Two shapes are congruent when they are the same shape and the same size, so one could be laid exactly on top of the other. In terms of moves: two shapes are congruent if some sequence of translations, reflections and rotations carries one onto the other. Those three moves never change lengths or angles, so they are called rigid motions. A triangle and its image after a rotation are always congruent.

Two shapes are similar when they have the same shape but maybe a different size. In terms of moves: two shapes are similar if rigid motions plus a dilation carry one onto the other. A triangle with sides 3, 4 and 5 and a triangle with sides 6, 8 and 10 are similar. Divide the matching sides: 6 ÷ 3 = 2, 8 ÷ 4 = 2, 10 ÷ 5 = 2. The scale factor is 2 for every pair, and the angles match.

To test for similarity, divide every pair of matching sides and see if you get the same number each time. A 2 by 3 rectangle and a 4 by 5 rectangle are not similar, because 4 ÷ 2 = 2 but 5 ÷ 3 is not 2. The second one is stretched more one way than the other. A 2 by 3 rectangle and a 4 by 6 rectangle are similar, with scale factor 2.

Every congruent pair is also similar, with scale factor 1. But similar shapes are not always congruent. A student who says two triangles are congruent just because their angles match has forgotten about size. Angles alone prove similarity; you need matching side lengths to prove congruence.

In a right triangle, the two sides that form the right angle are the legs. The third side, across from the right angle, is the hypotenuse, and it is always the longest. Call the legs a and b and the hypotenuse c. The Pythagorean theorem says a² + b² = c². For the 3-4-5 triangle, the legs are 3 and 4, so 9 + 16 = 25, and 25 = 5². It works.

Here is a proof by areas. Draw a big square with sides of length a + b. Inside it, place four copies of the right triangle, one in each corner, each with its legs along the edges. The four hypotenuses form a tilted square in the middle with side c. Now count the area two ways. The big square has area (a + b)², which multiplies out to a² + 2ab + b². The four triangles together have area 4 × (1/2 × a × b) = 2ab, and the tilted square has area c². So a² + 2ab + b² = 2ab + c². Take 2ab from both sides: a² + b² = c².

Use it to find a hypotenuse. Legs 5 and 12: 5² + 12² = 25 + 144 = 169. The hypotenuse is the number that squares to 169, which is 13, because 13 × 13 = 169. Legs 8 and 15: 64 + 225 = 289 = 17², so the hypotenuse is 17. Legs 6 and 8: 36 + 64 = 100, so 10.

The classic mistake is adding the legs instead of their squares: 5 + 12 = 17 is wrong; the hypotenuse is 13. Another is forgetting to take the square root at the end and answering 169. A third is using the theorem on a triangle that has no right angle, where it simply does not apply.

The theorem finds any one side from the other two. If the hypotenuse is missing, add: c² = a² + b². If a leg is missing, subtract: a² = c² − b². The hypotenuse is always the biggest, so its square goes alone on one side of the equation.

A right triangle has hypotenuse 10 and one leg 6. The other leg: 10² − 6² = 100 − 36 = 64, and √64 = 8. Check: 6² + 8² = 36 + 64 = 100 = 10². Another: hypotenuse 15, leg 9. Then 225 − 81 = 144, and √144 = 12. A student who adds instead of subtracting gets 225 + 81 = 306 and √306, which is about 17.5, longer than the hypotenuse. That is impossible, so an answer longer than the hypotenuse is a signal to go back.

Not every answer is a whole number. Legs 2 and 3 give 4 + 9 = 13, and √13 is not a whole number. It is between √9 = 3 and √16 = 4, closer to 3.6, because 3.6 × 3.6 = 12.96. A calculator gives 3.606. Write the exact answer as √13 and the estimate as about 3.6.

Two habits protect you. First, label the hypotenuse before you write anything, so you know whether to add or subtract. Second, check your result by putting all three sides back into a² + b² = c². If the check fails, the error is usually a squared number or a sign.

The theorem runs one way: a right triangle gives a² + b² = c². The converse runs the other way: if the three sides of a triangle satisfy a² + b² = c², with c the longest, then the angle across from c is a right angle. That is why the 3-4-5 rope works. 9 + 16 = 25, so the corner between the 3 and the 4 must be square.

To test a triangle, square the two shorter sides and add them, then compare with the square of the longest side. Sides 7, 24 and 25: 49 + 576 = 625, and 25² = 625. Equal, so it is a right triangle. Sides 4, 5 and 6: 16 + 25 = 41, but 6² = 36. Not equal, so no right angle. Sides 9, 12 and 15: 81 + 144 = 225 = 15². Right triangle.

The comparison also tells you what kind of triangle you have. If a² + b² is greater than c², the largest angle is less than 90 degrees, and the triangle is acute. If a² + b² is less than c², the largest angle is more than 90 degrees, and the triangle is obtuse. For 4, 5 and 6, 41 is more than 36, so the triangle is acute.

Builders use the converse every day. To check that a wall meets the floor at a right angle, mark 3 feet up the wall and 4 feet along the floor, then measure between the marks. Exactly 5 feet means square. More than 5 means the corner is too open; less means it is too tight. The common error is squaring the wrong side: c must be the longest, always.

Two points on a grid are usually not on the same row or column, so you cannot just count squares between them. But you can build a right triangle. From the first point go straight across, then straight up or down to the second point. The across distance and the up distance are the legs. The straight line between the points is the hypotenuse.

Find the distance from (1, 2) to (4, 6). Across: from x = 1 to x = 4 is 3. Up: from y = 2 to y = 6 is 4. Legs 3 and 4 make hypotenuse 5, so the distance is 5. Another: from (2, 1) to (7, 13). Across 7 − 2 = 5. Up 13 − 1 = 12. Distance √(25 + 144) = √169 = 13.

Negative coordinates work the same way; just be careful with the subtraction. From (−3, 2) to (3, 10): across is 3 − (−3) = 6, and up is 10 − 2 = 8. Distance √(36 + 64) = √100 = 10. If the answer is not a perfect square, estimate it. From (0, 0) to (2, 3): √(4 + 9) = √13, about 3.6.

The most common error is subtracting the wrong pairs, mixing an x with a y. Always subtract x from x and y from y. A second error is skipping the squares and adding the legs, which gives the walking distance along the grid lines, not the straight-line distance. The straight line is always shorter.

A ladder leaning on a wall makes a right triangle: the wall is one leg, the ground is the other, and the ladder is the hypotenuse. A 13-foot ladder has its base 5 feet from the wall. How high does it reach? 13² − 5² = 169 − 25 = 144, and √144 = 12. It reaches 12 feet. A 10-foot ladder with its base 6 feet out reaches √(100 − 36) = √64 = 8 feet.

Television screens are sold by their diagonal, corner to corner. A screen 40 inches wide and 30 inches tall has diagonal √(1600 + 900) = √2500 = 50 inches. A widescreen 48 inches wide and 27 inches tall has diagonal √(2304 + 729) = √3033, about 55 inches, since 55² = 3025. That is why a ’55-inch’ TV is much less than 55 inches wide.

Diagonals also save steps. A rectangular park is 300 m by 400 m. Walking around two sides from one corner to the opposite corner is 300 + 400 = 700 m. Cutting straight across the diagonal is √(90,000 + 160,000) = √250,000 = 500 m. The shortcut saves 200 m. The same triangle at a different scale tells you whether a 3-meter pole fits diagonally in a 2 m by 2 m box: √8 is about 2.8, so it does not.

The skill is spotting the right angle. Walls meet floors at right angles. Screens and doors have square corners. North and east are perpendicular directions. Once you see the square corner, label the two legs and the hypotenuse, decide which one is missing, and then add or subtract the squares.