GCSE · Maths · AQA · Spec 8300

Area and volume formulae — triangles, parallelograms, trapezia, prisms, cylinders

Drag a triangle's top corner sideways. It stretches and leans right over, yet its area doesn't change at all. That one fact unlocks every formula here.

Area · the height that matters

Slide the top corner. Watch the area.
8.0 cm5.0 cm90°5.4 cm7.8 cm20.0 cm²ABC

Base AB 8.0 cm. Perpendicular height 5.0 cm. Height meets base line at 90°. Side AC 5.4 cm. Side BC 7.8 cm. Area of triangle ABC 20.0 cm². Relationship: C can go anywhere on the line: the base and the perpendicular height stay the same, so the area stays the same. The sloping sides don't come into it.

Base AB8.0 cmPerpendicular height5.0 cmHeight meets base line at90°Side AC5.4 cmSide BC7.8 cmArea of triangle ABC20.0 cm²drag C ↔

C can go anywhere on the line: the base and the perpendicular height stay the same, so the area stays the same. The sloping sides don't come into it.

Drag corner C left or right along the line parallel to the base. The sides stretch and shrink. Before you let go, predict: what will the area do?

Watch out: The sides changed, but the area didn't budge. So a sloping side is never the height. The height is the straight-up distance from the base line to the top corner, meeting the base line at 90°. Drag C past B and the height lands outside the triangle. It still counts. (Park C right above A: side AC now stands at 90°, so just this once it is the height.)

Where the area formulae come from

?

Reason it through

How do four area formulae grow out of one rectangle?

Link 1 of 4

First link · your turn

Start simple. A rectangle is 6 cm long and 4 cm high. How would you find its area if you'd never seen a formula?

2
Locked — reveal the link above first
3
Locked — reveal the link above first
4
Locked — reveal the link above first

Spot the slip

This answer loses marks. Where does it go wrong?

A trapezium has parallel sides of 7 cm and 11 cm. Its perpendicular height is 5 cm, and one of its sloping sides is 6 cm. Work out its area.

A student's answer — which line goes wrong?

Volume · what counts as a prism

Prism or not?

Sort each solid. Is it a right prism, or not a prism?

Still to sort

Right prism (0)

The same cross-section all along its length, with ends at right angles to the length.

Where the line is: Every straight slice across it is the same shape and the same size.

Not a prism (0)

The slices change as you move along the solid.

Where the line is: If a slice gets bigger, smaller or changes shape as you move along, it isn't a prism, however neat the solid looks.

8 of 8 still to sort.

Imagine slicing each solid like a loaf of bread, straight across. Does every slice come out the same?

Predict, then check

Think of a prism as a stack of identical slices.

A slice of a prism is 1 cm thick, and its cross-section has an area of 15 cm². So the slice holds 15 cm³. Stack 8 of these identical slices to make a prism 8 cm long. What is the volume of the prism?

Your turn

Finish the cylinder

A cylindrical tin has a diameter of 10 cm and a height of 12 cm. Find its volume, correct to 1 decimal place.

  1. The tin is a prism: a circle stacked 12 cm high. So V = πr²h.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Three simple moves build every formula here: push a rectangle sideways, cut the result in half, stack a shape up. No list to memorise, just one idea you can rebuild.

What you need to know

  • Area of a triangle = ½ × base × perpendicular height. Area of a parallelogram = base × perpendicular height. Area of a trapezium = ½(a + b)h, where a and b are the parallel sides.
  • The height in every area formula is the perpendicular height: the distance at right angles to the base, never a sloping side.
  • A right prism has the same cross-section all along its length. Its volume = area of cross-section × length. Cuboids and cylinders are prisms, so V = length × width × height and V = πr²h.

The big picture

Every formula here grows from one idea. A rectangle's area counts unit squares. Push a parallelogram sideways (a shear) and it becomes a rectangle on the same base and perpendicular height, so its area is base × perpendicular height. A triangle is half that parallelogram: ½ × base × perpendicular height. Two trapezia make a parallelogram on base a + b, so a trapezium is ½(a + b)h. A right prism keeps the same cross-section all along, so its volume is area of cross-section × length; cuboids and cylinders (V = πr²h) are prisms too. The height is always perpendicular, never a sloping side.

Key points

1A rectangle's area counts unit squares: base × height.
2Shearing (pushing a shape sideways along a line parallel to its base) changes the sloping sides but not the area, so a parallelogram's area = base × perpendicular height.
3A triangle is half of the parallelogram on the same base and perpendicular height: area = ½ × base × perpendicular height.
4Two identical trapezia make a parallelogram on base a + b, so a trapezium's area = ½(a + b)h, with h the perpendicular distance between the parallel sides.
5A right prism has the same cross-section all along its length, with ends at right angles to its length. Pyramids, cones and spheres are not prisms.
6Volume of a prism = area of cross-section × length, because the cross-section is stacked along the length.
7A cuboid is a prism with a rectangular cross-section (V = length × width × height). A cylinder is a prism with a circular cross-section (V = πr²h, with r the radius, half the diameter).
8Areas are in square units such as cm², volumes in cubic units such as cm³.

Worked example

Problem

A water trough is a prism. Its end is a trapezium with parallel sides 30 cm and 50 cm and a perpendicular height of 20 cm. The trough is 120 cm long. Find its volume.

⚠ Watch out

Using a sloping side as the height. Every area formula needs the perpendicular height, the distance at right angles to the base. The other favourites: forgetting the ½ for a triangle or trapezium, and using the diameter instead of the radius for a cylinder.

🧠

Memory hook

Push it, halve it, stack it. Push a rectangle sideways: parallelogram. Halve it: triangle. Stack any cross-section along a length: prism. And the height is always the one that stands up straight.

✓

Check yourself

Cover the page. A triangular prism is 20 cm long. Its end has base 6 cm, perpendicular height 5 cm and sloping sides of about 5.8 cm. Find its volume. Which length didn't you need?

Flashcards

(13)
Area of a triangle?
½ × base × perpendicular height.
Area of a parallelogram?
Base × perpendicular height, the same as the rectangle it shears into.
Area of a trapezium, and what are a, b and h?
½(a + b)h. a and b are the two parallel sides; h is the perpendicular distance between them.
What is the 'perpendicular height' of a shape?
The distance from the base to the opposite side or corner, measured at right angles (90°) to the base. It can lie outside the shape.
Why doesn't pushing a shape sideways (a shear) change its area?
The base and the perpendicular height stay the same. Only the sloping sides change, and they aren't in the formula.
Why is the triangle formula 'half' of something?
Two identical triangles fit together to make a parallelogram on the same base and height, so one triangle is half of base × perpendicular height.
Why does the trapezium formula add a and b?
Two identical trapezia, one flipped, make a parallelogram whose base is a + b. One trapezium is half of it.
What makes a solid a right prism?
It has the same cross-section all along its length, with ends at right angles to the length.
Is a cone a prism?
No. Its circular slices shrink towards the tip, so the cross-section isn't the same all along.
A prism is a stack of identical slices. How does that give its volume?
A slice 1 unit thick holds as many unit cubes as its end has unit squares. Multiply by the number of slices (the length): V = cross-section area × length.
Why is a cuboid's volume length × width × height?
It's a prism with a rectangular cross-section: length × width, stacked up the height.
Volume of a cylinder, and what is r?
V = πr²h. r is the radius of the circular cross-section, half the diameter.
What units go with area, and with volume?
Area: square units such as cm². Volume: cubic units such as cm³.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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