Unit 21 · Geometry: Similarity, Trigonometry and Circles

Zoom in on a photo on your phone. The picture gets bigger, but the people in it do not become fatter or taller in a strange way. Every part grows by the same amount. That is a dilation. A dilation has a center (the point that stays put) and a scale factor, a number that tells how much every distance from the center gets multiplied.

Try it with coordinates. Put the center at the origin (0, 0) and use scale factor 3. The point (2, 3) moves to (3 × 2, 3 × 3) = (6, 9). The point (−1, 4) moves to (−3, 12). Each new point sits on the same line through the origin as the old one, just 3 times farther out. If the scale factor is 1/2, the point (6, 9) moves to (3, 4.5), which is half as far from the center.

A scale factor greater than 1 makes an enlargement. A scale factor between 0 and 1 makes a reduction. A common mistake is to add the scale factor instead of multiplying: (2, 3) with scale factor 3 is not (5, 6). Another is to dilate only the x-coordinate. Both coordinates get multiplied.

Two things happen under any dilation. Every length in the shape is multiplied by the scale factor. Every angle stays exactly the same. A triangle with a 40° angle still has a 40° angle after you triple it. That second fact is the key to the whole chapter: dilations keep shape and change only size.

A model of the Willis Tower on a shelf and the real tower on Wacker Drive have the same shape. Every angle on the model matches an angle on the tower, and every length on the model is the real length times the same tiny number. Figures like that are similar. Formally, two figures are similar if a dilation, possibly combined with a slide, a turn or a flip, carries one onto the other.

Check it with numbers. Triangle A has sides 3, 4 and 5. Triangle B has sides 6, 8 and 10. Pair up the corresponding sides, shortest with shortest and so on: 6/3 = 2, 8/4 = 2, 10/5 = 2. Every ratio is the same, so the sides are proportional and the scale factor from A to B is 2. Now look at triangle C with sides 6, 8 and 11. The ratios 2, 2 and 2.2 are not all equal, so C is not similar to A.

Perimeter and area scale differently. Triangle A has perimeter 12; triangle B has perimeter 24, twice as much. But the area of A is 1/2 × 3 × 4 = 6, and the area of B is 1/2 × 6 × 8 = 24, four times as much. Length is multiplied by the scale factor k. Area is multiplied by k², because area has two dimensions and each one is stretched. A common mistake is to double the area when the sides double.

Checking three angles and three side ratios every time is slow. For triangles there are shortcuts. The most useful is AA similarity: if two angles of one triangle equal two angles of another, the triangles are similar. Why does that work? The three angles of a triangle always add to 180°, so once two angles match, the third must match too, and the shapes are the same.

Try it. One triangle has angles 40° and 60°. Another has angles 60° and 80°. The first triangle’s third angle is 180 − 40 − 60 = 80°. So the first has angles 40°, 60°, 80° and the second has 60°, 80° and 180 − 60 − 80 = 40°. Same three angles, so they are similar by AA, even though nobody told you a single side length.

Two more shortcuts use sides. SAS similarity: two pairs of sides are proportional and the angles between those sides are equal. A triangle with sides 4 and 6 around a 50° angle is similar to one with sides 6 and 9 around a 50° angle, because 6/4 = 1.5 and 9/6 = 1.5. SSS similarity: all three pairs of sides are proportional. Sides 2, 3, 4 and sides 4, 6, 8 give ratios 2, 2, 2, so those triangles are similar.

A common mistake with SAS is using an angle that is not between the two sides. The angle must be the one the two sides form. Another mistake is thinking AA needs the angles to be in the same order around the triangle. It does not; any two matching angles will do.

Picture a triangle-shaped garden bed with one straight path cutting across it, parallel to the back edge. The path splits each of the two other edges into two pieces. Those pieces come in the same ratio on both edges. This is the side-splitter rule: a line parallel to one side of a triangle divides the other two sides proportionally.

Here is why. Call the triangle ABC and let the parallel line meet side AB at D and side AC at E. Because DE is parallel to BC, the angle at D equals the angle at B (corresponding angles from parallel lines). Both triangles share angle A. So triangle ADE is similar to triangle ABC by AA. Similar triangles have proportional sides, and a little algebra turns that into AD/DB = AE/EC.

Use it. Suppose AD = 4, DB = 6 and AE = 6. Set up 4/6 = 6/EC. Cross-multiply: 4 × EC = 36, so EC = 9. Check with the whole sides: AB = 10 and AC = 15, and 4/10 = 0.4 while 6/15 = 0.4. The pieces match. A common mistake is to compare a piece with the wrong whole, like 4/10 = 6/9. Keep pieces with pieces or wholes with wholes.

A special case: when D and E are midpoints of their sides, the segment DE is a midsegment. It is parallel to the third side and exactly half as long. If BC is 14, then DE is 7. Builders use this when they brace a triangular roof truss halfway up.

You have used a² + b² = c² since middle school. Similar triangles show why it is true. Start with a right triangle with legs a and b and hypotenuse c. From the right angle, draw the altitude straight down to the hypotenuse, hitting it at a right angle. The hypotenuse is now split into two pieces: d next to leg a, and e next to leg b, with d + e = c.

The altitude makes two small right triangles. Each small triangle shares an acute angle with the big one, and each has a right angle, so each is similar to the big triangle by AA. Compare the triangle containing leg a with the big triangle: a/c = d/a, which gives a² = c × d. Compare the other small triangle: b/c = e/b, so b² = c × e. Add them: a² + b² = c × d + c × e = c(d + e) = c × c = c².

Check it on a 6-8-10 triangle. Here a = 6, b = 8, c = 10. Then d = a²/c = 36/10 = 3.6 and e = b²/c = 64/10 = 6.4, and sure enough 3.6 + 6.4 = 10. The altitude itself satisfies h² = d × e = 3.6 × 6.4 = 23.04, so h = 4.8. A second route: the altitude equals ab/c = 48/10 = 4.8. Both routes agree.

The theorem also runs in reverse. If a triangle’s sides satisfy a² + b² = c², it is a right triangle. Sides 5, 12, 13 work because 25 + 144 = 169. Sides 4, 5, 6 do not, since 16 + 25 = 41 and 36 ≠ 41, so that triangle has no right angle. A common mistake is adding the legs instead of their squares: 6 + 8 is not 10.

Cut a square along its diagonal and you get two 45-45-90 triangles. Both legs are equal. If each leg is 1, the hypotenuse is √(1 + 1) = √2, about 1.414. Since every 45-45-90 triangle is similar to this one, the rule is: hypotenuse = leg × √2. A square tile 10 inches on a side has a diagonal of 10√2, about 14.1 inches.

Cut an equilateral triangle in half from a corner to the middle of the opposite side and you get a 30-60-90 triangle. If the equilateral triangle has sides 2, the half-triangle has hypotenuse 2 and short leg 1 (half the base). The long leg is √(4 − 1) = √3, about 1.732. The rule: short leg x, hypotenuse 2x, long leg x√3. The short leg is always across from the 30° angle.

Try one. A 30-60-90 triangle has short leg 4. The hypotenuse is 2 × 4 = 8. The long leg is 4√3, about 6.93. Check with the Pythagorean theorem: 4² + (4√3)² = 16 + 48 = 64 = 8². Now go backward: if the hypotenuse is 12, the short leg is 12 ÷ 2 = 6 and the long leg is 6√3, about 10.4.

Common mistakes: putting √3 on the hypotenuse instead of the long leg, and forgetting that the short leg is across from the smallest angle. For a 45-45-90 triangle, if you know the hypotenuse, divide by √2 to get a leg: hypotenuse 10 gives legs of 10/√2 = 5√2, about 7.07.

Every right triangle with a 30° angle is similar to every other one, by AA (they share 30° and 90°). Similar means proportional sides, so the ratio of any two sides is the same in all of them. That is a powerful fact. It means a ratio like (side across from 30°) ÷ (hypotenuse) is a fixed number that belongs to 30°, not to any one triangle. Ratios like this are the tools of trigonometry.

Pick an acute angle A in a right triangle. The side across from A is the opposite side. The leg next to A is the adjacent side. The longest side is the hypotenuse. Three ratios get names: sine of A = opposite ÷ hypotenuse, cosine of A = adjacent ÷ hypotenuse, tangent of A = opposite ÷ adjacent. Many students remember them with SOH-CAH-TOA.

Work through a 3-4-5 triangle. Let angle A be across from the side of length 3. Then opposite = 3, adjacent = 4, hypotenuse = 5. So sin A = 3/5 = 0.6, cos A = 4/5 = 0.8, tan A = 3/4 = 0.75. Now look at the other acute angle B, across from 4: sin B = 4/5 = 0.8, cos B = 3/5 = 0.6. The sine of one acute angle equals the cosine of the other, because A and B add to 90° and the opposite side of one is the adjacent side of the other.

A common mistake is labeling sides before choosing the angle. Which leg is opposite and which is adjacent changes when you switch angles. The hypotenuse never changes. Another mistake is writing a ratio bigger than 1 for sine or cosine; the hypotenuse is the longest side, so those two ratios are always less than 1.

A 20-foot ladder leans against a wall at a 70° angle with the ground. How high up the wall does it reach? Draw the right triangle. The ladder is the hypotenuse (20 ft). The height is the side opposite the 70° angle. Opposite and hypotenuse means sine: sin 70° = height ÷ 20. A calculator gives sin 70° ≈ 0.940, so height = 20 × 0.940 = 18.8 feet.

The steps are always the same. First, mark the angle you know and label the two sides involved as opposite, adjacent or hypotenuse. Second, pick the ratio that uses exactly those two sides. Third, write the equation and solve. If the unknown is on top of the fraction, multiply. If it is on the bottom, divide.

Two more examples. A roof rises at 35° and the horizontal run is 50 feet. The rise is opposite, the run is adjacent, so tangent: tan 35° ≈ 0.700, and rise = 50 × 0.700 = 35.0 feet. Now suppose a zip line drops 12 meters and makes a 40° angle with the ground. The 12 m is opposite the angle and the cable is the hypotenuse, so sin 40° = 12 ÷ cable. Then cable = 12 ÷ sin 40° = 12 ÷ 0.643 ≈ 18.7 meters.

The most common mistake is choosing the wrong ratio, usually by mislabeling opposite and adjacent. Always label from the angle you are using. The second most common is multiplying when you should divide. When the unknown is the hypotenuse under a sine or cosine, you divide the known side by the ratio, and the answer must come out bigger than the known side.

Sometimes the sides are easy to measure and the angle is the mystery. Building codes under the Americans with Disabilities Act say a wheelchair ramp may rise at most 1 unit for every 12 units of run. What angle is that? The rise is opposite the angle and the run is adjacent, so tan θ = 1/12 ≈ 0.0833. To go from a ratio back to the angle, use the inverse tangent, written tan⁻¹. A calculator gives tan⁻¹(0.0833) ≈ 4.8°. A legal ramp is gentle.

The three inverse functions undo the three ratios. If sin θ = 0.6, then θ = sin⁻¹(0.6) ≈ 36.9°. If cos θ = 0.6, then θ = cos⁻¹(0.6) ≈ 53.1°. Notice these two add to 90°, which makes sense: in a 3-4-5 triangle, the angle with sine 0.6 and the angle with cosine 0.6 are the two acute angles. Once you find one acute angle, the other is 90° minus it.

Work an example. A roof rises 6 feet over a run of 9 feet. What is the roof’s angle? tan θ = 6/9 ≈ 0.667, so θ = tan⁻¹(0.667) ≈ 33.7°. Check by going forward: tan 33.7° ≈ 0.667, and 9 × 0.667 ≈ 6. The other angle of the triangle is 90 − 33.7 = 56.3°.

A common mistake is typing sin⁻¹ when you built a tangent ratio; the inverse must match the ratio you formed. Another is writing 1/sin instead of sin⁻¹; the little −1 means "inverse function," not "reciprocal." And keep your calculator in degree mode, or the answer will come out in a different unit you will meet in the next chapter.

Stand on the lakefront and look up at the top of a skyscraper. Your line of sight tilts up from the horizontal. That tilt is the angle of elevation. Now imagine standing on the roof looking down at a boat. Your line of sight tilts down from horizontal. That is the angle of depression. Because the ground and the horizontal line from the roof are parallel, the angle of depression from the roof equals the angle of elevation from the boat. They are alternate interior angles.

Elevation example. You stand 40 meters from a building, and your eyes are 1.5 meters above the ground. The angle of elevation to the roof is 60°. The height above your eyes is opposite the angle and 40 m is adjacent, so use tangent: height = 40 × tan 60° = 40 × 1.732 = 69.3 m. Add your eye height: 69.3 + 1.5 = 70.8 m. Forgetting the eye height is the most common mistake here.

Depression example. A lighthouse keeper 100 meters above the water sees a boat at an angle of depression of 20°. The angle of elevation from the boat is also 20°. From the boat, the 100 m tower is opposite and the distance across the water is adjacent, so tan 20° = 100 ÷ distance. Then distance = 100 ÷ tan 20° = 100 ÷ 0.364 ≈ 274.7 meters.

A drawing error causes more wrong answers than arithmetic does. The angle of depression is measured from the horizontal line at the top, not from the vertical wall. If you put the 20° next to the tower instead of next to the horizontal, you will solve for the wrong side. Draw the horizontal line first, then the angle.

The oldest trick in surveying is the shadow method. On a sunny afternoon, a 1.8-meter-tall person casts a 2.4-meter shadow. At the same moment, a nearby tree casts a 12-meter shadow. The Sun’s rays are parallel, so both triangles have the same angle at the tip of the shadow, and both have a right angle at the ground. AA similarity again. Then tree height ÷ 12 = 1.8 ÷ 2.4, and the tree is 12 × 1.8 ÷ 2.4 = 12 × 0.75 = 9 meters tall. Measure both shadows at the same time; a few minutes later the Sun has moved.

The mirror method from the chapter story works when the Sun is not out. A mirror on the ground reflects light at the same angle it arrives, so the triangle from your eye to the mirror and the triangle from the object’s top to the mirror share an angle at the mirror and a right angle at the ground. With eye height 1.6 m, 2.5 m from you to the mirror and 30 m from the mirror to a building, the building’s height is 1.6 × 30 ÷ 2.5 = 19.2 m.

You can measure across a river the same way. Stand directly across from a tree on the far bank. Walk 40 m along your bank and drive in a stake. Walk 10 m farther. Then turn and walk straight away from the river until the stake lines up with the tree; say that takes 8 m. The small triangle (10 by 8) is similar to the big one (40 by the river’s width). River width = 40 × 8 ÷ 10 = 32 m.

Every method rests on the same two facts: two triangles with two matching angles are similar, and similar triangles have proportional sides. The common mistake is pairing sides that do not correspond, like the person’s height with the tree’s shadow. Write the proportion as height over shadow for both, or small over big for both, and keep it consistent.

A pizza is cut into 8 equal slices. Every cut goes through the center. The point of each slice, at the center, forms a central angle. Since the eight angles fill the whole circle, and a full turn is 360°, each central angle is 360 ÷ 8 = 45°. The curved crust of a slice is an arc. We measure an arc in degrees too, and an arc’s measure equals its central angle. One slice’s crust is a 45° arc, and three slices together make a 135° arc.

Some vocabulary. A radius runs from the center to the circle. A chord is a segment with both ends on the circle. A diameter is a chord through the center; it is the longest chord and equals two radii. Two points on a circle split it into two arcs. The smaller one, less than 180°, is the minor arc; the larger one is the major arc. Their measures add to 360°. If a minor arc is 110°, the major arc is 360 − 110 = 250°.

A clock is a circle with 12 equal parts, so each hour mark is 360 ÷ 12 = 30° from the next. From 12 to 4 the minute hand turns through 4 × 30 = 120°. From 12 to 9 it turns 270°, and the minor arc between 12 and 9 is only 90°. A common mistake is to think the arc’s length in inches matters here. Arc measure in degrees is about the angle at the center, not the size of the circle. A 45° arc on a tiny circle and a 45° arc on a giant one have the same degree measure.

Move the vertex of the angle from the center out to the circle itself. Now you have an inscribed angle, with its vertex on the circle and its two sides as chords. The arc between the sides is the intercepted arc. Here is the surprising rule: an inscribed angle is half the measure of its intercepted arc. If the arc is 80°, the inscribed angle is 40°. A central angle on that same arc would be the full 80°.

Why half? Take the simplest case, where one side of the inscribed angle is a diameter. Draw a radius to the far end of the other side. That radius and the diameter’s radius are equal, so they make an isosceles triangle with two equal base angles, each equal to the inscribed angle. The central angle is an exterior angle of that triangle, so it equals the sum of the two equal angles, which is twice the inscribed angle. The other cases follow by adding or subtracting this one.

Three facts follow. First, every inscribed angle that opens onto the same arc has the same measure, since they are all half of the same arc. Second, an inscribed angle whose sides pass through the ends of a diameter intercepts a 180° arc, so it measures 90°. This is often called Thales’ theorem. Third, if a four-sided figure has all four corners on a circle, its opposite angles add to 180°, because together they intercept the whole 360° circle.

Work one. An inscribed angle measures 35°. Its intercepted arc is 2 × 35 = 70°. If another inscribed angle sits anywhere else on the circle and opens onto that same 70° arc, it is also 35°. The common mistake is to forget which angle is which: central angles equal the arc, inscribed angles are half of it.

A bicycle wheel sits on flat ground. The ground touches the tire at exactly one point. A line that touches a circle at only one point is a tangent line, and the touching point is the point of tangency. The spoke from the hub straight down to that point is a radius, and it makes a right angle with the ground. That is always true: a tangent is perpendicular to the radius drawn to the point of tangency.

That right angle lets you use the Pythagorean theorem. Suppose a point P is 10 cm from the center of a circle of radius 6 cm, and you draw the tangent from P. The radius (6), the tangent segment, and the segment from P to the center (10) form a right triangle with the right angle at the point of tangency. Tangent length = √(10² − 6²) = √(100 − 36) = √64 = 8 cm. From any outside point you can draw two tangents, and their lengths are equal.

A line that cuts through a circle at two points is a secant, and the piece inside is a chord. If you draw a segment from the center perpendicular to a chord, it cuts the chord exactly in half. So the center, the midpoint of the chord and one end of the chord form a right triangle. Example: a circle has radius 13 and a chord sits 5 units from the center. Half the chord is √(13² − 5²) = √(169 − 25) = √144 = 12, so the whole chord is 24.

The common mistake is to build the right triangle with the wrong sides. The radius to the point of tangency is a leg, not the hypotenuse; the segment from the outside point to the center is the hypotenuse. For chords, the distance from the center reaches the middle of the chord, so remember to double the leg you find.

Wrap a string around a can and then measure the string against the can’s width. The string is always a little more than 3 times the width, about 3.14 times. That ratio is the same for every circle, and it has a name: π (pi), roughly 3.14159. So the distance around a circle, its circumference, is C = π × d, or C = 2πr since the diameter is twice the radius. A circle with radius 7 cm has circumference 2 × π × 7 = 14π ≈ 44.0 cm.

Area takes a trick. Cut a circle into many thin wedges, like pizza slices, and lay them side by side, points alternating up and down. They form a shape close to a parallelogram. Its height is the radius r. Its base is half the crust, half the circumference, which is πr. The area of a parallelogram is base × height, so the area of the circle is πr × r = πr². With more, thinner wedges, the shape gets closer to a true parallelogram, and the formula becomes exact.

Try radius 7 again: area = π × 7² = 49π ≈ 153.9 square cm. Now a circle with diameter 12: first find the radius, 12 ÷ 2 = 6, then area = 36π ≈ 113.1. The most common mistake is to square the diameter instead of the radius, which gives an answer 4 times too big. The second most common is to mix up the two formulas; circumference uses r once (a length), area uses r twice (a length times a length).

A slice of a circle is a sector, and its curved edge is an arc. If the slice’s central angle is θ degrees, the slice is θ/360 of the whole circle. So arc length = (θ/360) × 2πr and sector area = (θ/360) × πr². One fraction handles both, and the fraction is the same whether the circle is a coin or a running track.

Work it. A circle has radius 9 and a sector has a 60° angle. The fraction is 60/360 = 1/6. The full circumference is 2π × 9 = 18π, so the arc length is 18π ÷ 6 = 3π ≈ 9.42. The full area is π × 81 = 81π, so the sector area is 81π ÷ 6 = 13.5π ≈ 42.4. Both answers are one sixth of the whole, which is a good way to check.

Real example: a pizza with radius 8 inches is cut into 6 equal slices. Each slice is 1/6 of the pizza. The whole pizza’s area is 64π ≈ 201.1 square inches, so one slice is 201.1 ÷ 6 ≈ 33.5 square inches. The crust on one slice, its arc, is 1/6 of 16π ≈ 50.3 inches, about 8.4 inches.

Two common mistakes. One is confusing arc length (a distance, measured in inches) with arc measure (an angle, measured in degrees); a 60° arc on a big pizza is longer than a 60° arc on a small one, even though both measure 60°. The other is forgetting the fraction and reporting the whole circumference or whole area.

Degrees are an ancient choice; 360 was handy for old calendars, not for mathematics. Here is a more natural unit. Take a circle and bend a piece of string exactly as long as the radius along the edge. The central angle that string covers is one radian. Because the whole circumference is 2πr, the radius fits around the circle 2π times, so a full turn is 2π radians, about 6.28. Half a turn, 180°, is π radians.

To convert, use 180° = π radians. Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. So 60° = 60 × π/180 = π/3 ≈ 1.047 radians, and π/2 radians = 90°. The quarter turn, 90°, is π/2; the eighth turn, 45°, is π/4; the twelfth turn, 30°, is π/6.

Radians make arc length simple. Since an angle of θ radians is θ/(2π) of a full turn, the arc is θ/(2π) × 2πr = rθ. No fraction over 360. With radius 9 and angle π/3, the arc is 9 × π/3 = 3π ≈ 9.42, matching the 60° arc from the last lesson. With radius 4 and angle 1.5 radians, the arc is 4 × 1.5 = 6 units, no π needed.

The common mistake is mixing units: putting degrees into s = rθ gives an absurd answer (60 × 9 = 540 for a circle of circumference 56.5). Convert first. Calculators have a radian mode for this; if sin 30 comes out as −0.988 instead of 0.5, the calculator is in radians.

A radio station’s tower sits at the origin of a map, and its signal reaches 50 miles. Which points are exactly 50 miles away? A point (x, y) is at distance √(x² + y²) from the origin, by the Pythagorean theorem. Setting that equal to 50 and squaring both sides gives x² + y² = 2500. That is the equation of a circle with center (0, 0) and radius 50. Every point on the circle makes the equation true, and no other point does.

Move the center to (h, k). The distance from (x, y) to (h, k) is √((x − h)² + (y − k)²). Set it equal to r and square: (x − h)² + (y − k)² = r². This is the standard form of a circle. For center (2, −3) and radius 5: (x − 2)² + (y − (−3))² = 25, which is (x − 2)² + (y + 3)² = 25. Watch the signs: a plus inside the parentheses means a negative coordinate.

Test a point. Is (5, 1) on that circle? Substitute: (5 − 2)² + (1 + 3)² = 3² + 4² = 9 + 16 = 25. Yes. Is (4, 0)? (4 − 2)² + (0 + 3)² = 4 + 9 = 13, which is less than 25, so (4, 0) is inside the circle. A point that gives more than 25, like (8, 0), is outside the circle.

Reading an equation backwards is just as important. From (x + 1)² + (y − 4)² = 9, the center is (−1, 4), flipping each sign, and the radius is √9 = 3, not 9. Two common mistakes: reading the center as (1, −4), and reporting r² as the radius. Always take the square root of the right side.

Sometimes a circle’s equation arrives multiplied out, like x² + y² − 6x + 4y − 12 = 0. The center and radius are hidden. To find them, rebuild the squares. Group the x terms and the y terms: (x² − 6x) + (y² + 4y) = 12. To make x² − 6x into a perfect square, take half of −6, which is −3, and square it: 9. Add 9 to both sides. For y² + 4y, half of 4 is 2, squared is 4; add 4 to both sides.

Now the left side factors: (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25. The center is (3, −2) and the radius is √25 = 5. This is completing the square, the same move used to solve quadratics. Check it by expanding: (x − 3)² = x² − 6x + 9 and (y + 2)² = y² + 4y + 4, so the left side is x² + y² − 6x + 4y + 13, and 13 − 25 = −12 matches the original.

Another way a circle can be described: by the two ends of a diameter. Suppose a diameter runs from (1, 2) to (7, 10). The center is the midpoint: ((1 + 7)/2, (2 + 10)/2) = (4, 6). The radius is half the diameter’s length. The diameter’s length is √((7 − 1)² + (10 − 2)²) = √(36 + 64) = √100 = 10, so r = 5 and the equation is (x − 4)² + (y − 6)² = 25.

The common mistake in completing the square is adding the new numbers to only one side of the equation. Whatever you add on the left, add on the right too. Another is halving the wrong coefficient; you halve the number in front of x (or y), never the number in front of x².

A can of soup is a cylinder: two circular ends and a curved side. Think of it as a stack of thin circular coins. Each coin has area πr², and there are enough of them to reach height h. So the volume is V = πr²h. A soup can with radius 4 cm and height 11 cm holds π × 16 × 11 = 176π ≈ 552.9 cubic centimeters, which is about 553 milliliters.

The same stacking idea gives the volume of any prism, a solid with two identical parallel ends and flat sides. Volume = (area of the base) × height, or V = Bh. A rectangular box 5 cm by 4 cm by 12 cm has base area 5 × 4 = 20 and height 12, so V = 240 cubic cm. A triangular prism with a base triangle of area 15 and length 8 has volume 15 × 8 = 120.

Now tilt the stack of coins so it leans like a fallen tower. Every coin is still there, and each still has the same area, so the volume has not changed. This is Cavalieri's principle: if two solids have the same height and the same cross-sectional area at every level, they have the same volume. It is why a slanted cylinder has the same formula as an upright one, as long as h is the straight-up height.

Doubling the radius of a cylinder multiplies the volume by 4, because the radius is squared; doubling the height only doubles the volume. The common mistake is squaring the diameter, which makes the answer 4 times too big. Units also trip people up: length in cm gives volume in cubic cm (cm³), and 1 cm³ equals 1 milliliter.

Fill a paper cone with water and pour it into a cylinder that has the same base and the same height. It fills one third. Do it three times and the cylinder is full. That experiment gives the rule: a cone has volume V = (1/3)πr²h, and a pyramid with base area B has V = (1/3)Bh. The one-third rule holds for every pyramid and cone, whatever the shape of the base.

Work an example. An ice cream cone has radius 3 cm and height 8 cm. The matching cylinder would hold π × 9 × 8 = 72π. The cone holds a third of that: 24π ≈ 75.4 cubic cm. A square pyramid with base 10 m by 10 m and height 12 m has base area 100, so V = (1/3) × 100 × 12 = 400 cubic meters.

Two heights matter for a cone. The height h goes straight from the tip down to the center of the base. The slant height runs along the outside from the tip to the edge of the base. They form a right triangle with the radius: r² + h² = (slant height)². A cone with radius 3 and height 4 has slant height 5. Volume uses h, never the slant height. Using the slant height by mistake makes the volume too large.

Why one third? Cavalieri’s principle and a slicing argument show it. A cube can be cut into three identical square pyramids that meet at the cube’s center of one face, so each pyramid is one third of the cube. Then any pyramid or cone with the same base area and height has the same cross-sections at every level, so the same one-third rule applies.

A sphere is the set of all points at distance r from a center, a ball. Its volume formula is V = (4/3)πr³. A ball with radius 3 cm holds (4/3) × π × 27 = 36π ≈ 113.1 cubic cm. A hemisphere, half a ball, holds half that: (2/3)πr³ = 18π ≈ 56.5 cubic cm for radius 3. Notice that r is cubed, so doubling the radius multiplies the volume by 8.

Where does 4/3 come from? Cavalieri’s principle. Put a hemisphere of radius r flat side down. Next to it put a cylinder of radius r and height r, and scoop out of it a cone, point down, that has the same radius and height. Now slice both solids at some height y above the table. The hemisphere’s slice is a circle with radius √(r² − y²), by the Pythagorean theorem, so its area is π(r² − y²). The cylinder’s slice is a full circle of area πr² with a hole of radius y (the cone gets wider going up), area πy². The ring’s area is πr² − πy².

The two slices have equal area at every height. By Cavalieri’s principle, the solids have equal volume. The cylinder minus the cone is πr² × r − (1/3)πr² × r = (2/3)πr³. That is the hemisphere. Double it for the whole sphere: (4/3)πr³. Archimedes proved this more than two thousand years ago and was so proud of it that he asked for a sphere and cylinder carved on his tombstone.

Try the check at r = 2 and height y = 1. Hemisphere slice: π(4 − 1) = 3π. Ring: π × 4 − π × 1 = 3π. Equal. The common mistake with spheres is using the surface area formula 4πr² for volume; that formula gives square units, and volume must be cubic units, with r cubed.

Two blocks the same size can have very different weights. The difference is density: mass divided by volume, usually in grams per cubic centimeter. Water is the reference at 1.0 g/cm³. A metal block 10 cm by 5 cm by 4 cm has volume 200 cm³. If it has mass 540 g, its density is 540 ÷ 200 = 2.7 g/cm³, which matches aluminum. Iron is about 7.9 g/cm³, so an iron block of the same size would have mass 7.9 × 200 = 1,580 g.

Real objects are rarely a single simple solid, so model them as combinations. A grain silo is a cylinder with a hemisphere on top. With radius 3 m and a cylinder height of 10 m: cylinder = π × 9 × 10 = 90π, hemisphere = (2/3) × π × 27 = 18π, total = 108π ≈ 339.3 cubic meters. A pencil is a cylinder plus a cone. A capsule is a cylinder plus two hemispheres, which together make one sphere.

Density also describes how crowded a region is. Population density is people divided by area. Chicago has roughly 2.7 million people in about 227 square miles, so its density is about 2,700,000 ÷ 227 ≈ 11,900 people per square mile, or about 12,000. A suburb with 40,000 people in 20 square miles has density 2,000 per square mile, six times less crowded.

The common mistake is dividing in the wrong order. Density is the amount per unit of space, so mass goes on top and volume on the bottom; an answer far less than 1 for a metal is a sign the division was flipped. In modeling, watch for double-counting: a cylinder with a hemisphere on top has no lid, so do not add the cylinder’s top circle to the surface area.