Every formula here is a rectangle in disguise. Parallelograms, triangles and trapezoids; shapes built from other shapes; the circle, around and inside; nets and surface area; and the volume of a prism — plus how to tell which one a problem is asking for. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Length is counted in units. Covering is area, counted in SQUARE units. Filling is volume, counted in CUBIC units. Decide which the question wants before you pick a formula — then let the units check you.
Four things people get wrong
⚠️People often think…
The slanted side of a triangle is its height.
Height meets the base at a right angle. The slanted side is longer, and using it gives an answer that is too big.
No right angle, not the height.
⚠️People often think…
A 12-inch pizza has r = 12.
A 12-inch pizza is 12 inches ACROSS. That is the diameter, so r = 6. Area uses the radius, never the diameter.
Across is diameter. Halve it first.
⚠️People often think…
πr² means multiply π by r, then square it.
Square the radius first, then multiply by π. With r = 6: 36, then 36 × 3.14 = 113.04. Doing it the other way gives about 355.
Only the r is squared.
⚠️People often think…
Surface area is the three faces you can see.
A box has six faces, and the hidden three match the three you see. Three faces is exactly half the answer.
Count six, or double the three.
Which question is it? Then do it
1Watch one
A box is 5 by 3 by 2. How much fits inside?
“Fits inside” means volume.
Volume is counted in cubic units.
5 × 3 × 2 = 30.
30 cubic units.
2Do one with me
A triangle has base 10 and height 6. Find the area.
The formula is ½ × base ×
½ × 10 × 6 =
The units are square or cubic?
💬One sentence, then you move on
How do you tell that a question wants area and not volume?
3Try one
A circle is 12 inches across. Find the area. Use π ≈ 3.14.
I want a hint first
12 across is the diameter, so the radius is 6. Square the 6 first, then multiply by 3.14.
💬Last one — then you're done here
Why does the word “across” change the whole problem?
Where this goes
Where this lives
Paint for a wall is area. Water in a tank is volume. Trim around a window is length. Buy the wrong one and you come home short.
What this feeds
Next unit is angles, triangles and scale drawings, where the picture itself hands you the equation.
Name one thing you would measure in square units, and one in cubic units.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a rectangle to everything else — the ladder
Rectangle (base × height) → parallelogram (slide the end: still base × height) → triangle (half of that) → trapezoid (average the bases) → circle (πr²: slices rearranged into a rectangle) → net (add the faces) → prism (base area × height).
Every formula here is a rectangle in disguise.
🤝 The whole topic in one idea
Length is counted in units, covering in SQUARE units, filling in CUBIC units. Decide which one the question wants before you reach for a formula — then let the units check your answer.
⚠️ Traps the test loves
The slanted side is not the height. Height meets the base at a right angle.
Forgetting the ½ on triangles and trapezoids.
Radius or diameter? Area uses the RADIUS. A 12-inch pizza has r = 6.
πr² means π × r × r — square first. It is not (π × r)², and it is not 2πr.
Surface area needs all six faces. Three faces is half the answer.
In a composite figure, the missing length is usually a subtraction.
📐 The formulas — the chart
Shape
Formula
Remember
Parallelogram
A = b × h
h is straight up, not the slanted side
Triangle
A = ½ × b × h
half of a parallelogram
Trapezoid
A = ½ × (b₁ + b₂) × h
the bases are the parallel sides
Circle — around
C = π × d = 2 × π × r
a length: plain units
Circle — inside
A = π × r²
radius, squared first
Rectangular prism — covering
SA = 2lw + 2lh + 2wh
three pairs of twin faces
Any prism — filling
V = B × h
B is the AREA of the base
🎯 How the test will ask
A shape with an extra number on it — pick the base and the HEIGHT, ignore the slanted side.
A composite figure — cut it, find the missing length, add the pieces (or subtract a hole).
A circle given by its diameter — halve it before finding the area.
A net — find each face, add them all.
A story — wrapping, painting, filling, fencing: name the measure first.
A box with fraction edges — multiply the mixed numbers carefully, two at a time.
✅ Before the test, can you…
Find the area of a parallelogram, a triangle and a trapezoid — and explain the ½?
Cut a composite figure into pieces and find its area?
Find a circle’s circumference and area from either the radius or the diameter?
Sketch a net and use it to find surface area?
Find the volume of a prism, even with fraction edges?
Tell from a story whether it wants length, area, surface area or volume?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: fill in a net, cut a plot into rectangles with your own hands and find the length nobody gave you, decide which measure a job needs, then settle a pizza argument with numbers. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.