Area and Perimeter: Formulas for Rectangles, Triangles and Trapeziums

The difference between area and perimeter, and where the formulas for rectangles, triangles, parallelograms and trapeziums (trapezoids) come from, all built from the rectangle.

3 min read Supports 4 lessons

Fencing a field needs its perimeter: the total length of its edge. Turfing the field needs its area: how much ground it covers. The two are quite different and are measured in different units.

A $4 \times 7$ rectangle: area $28$ square units, perimeter $4 + 7 + 4 + 7 = 22$ units.

Watch out

Area is in square units, like cm² or m². Perimeter is in plain units, like cm or m.

Rectangles first

A rectangle $b$ units long and $h$ units high is $h$ rows of $b$ unit squares, so its area is $bh$ . Every other formula on this page comes from rearranging shapes into rectangles.

Parallelograms

Slice a triangle off one end of a parallelogram and slide it to the other end. You get a rectangle with the same base and height, so the area is still $bh$ .

Triangles

Two copies of any triangle fit together into a parallelogram. So a triangle is half a parallelogram: $\frac12 bh$ .

Trapeziums

Two copies of a trapezium (trapezoid), one turned upside down, make a parallelogram whose base is $a + b$ . Halve it.

Area formulas

\text{rectangle } bh \quad \text{triangle } \tfrac12 bh \quad \text{parallelogram } bh \quad \text{trapezium } \tfrac12(a + b)h

Tip

The height is always measured perpendicular to the base, straight up at a right angle, not along a slanted side.

$\text{Trapezium: parallel sides } 5 \text{ and } 9, \text{ height } 4$
Find the area.

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Compound shapes

For an L-shape or any other combination, split it into rectangles and triangles, find each area, and add. Or find a bigger rectangle and subtract the pieces that are missing.

Worked example

A rectangular garden $10$ m by $6$ m has a square pond $2$ m by $2$ m. What area of lawn is left?

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Whole garden.
$10 \times 6 = 60 \text{ m}^2$

Scaling

Key idea

Doubling every length of a shape doubles its perimeter but multiplies its area by $4$ . Scaling lengths by $k$ scales area by $k^2$ .

The same thinking extends to curved shapes, giving the area of a circle, and into three dimensions, giving volume.

Common questions

What is the difference between area and perimeter?: Perimeter is the length around the outside of a shape. Area is the space inside it. Perimeter is in units such as m; area is in square units such as m².
Why is the area of a triangle half base times height?: Two identical triangles fit together to make a parallelogram with area base times height, so one triangle is half of that.

Practise area and perimeter

The Area & Perimeter lesson has worked examples and unlimited practice that adapts to you.

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Common questions

What is the difference between area and perimeter?

Perimeter is the length around the outside of a shape. Area is the space inside it. Perimeter is in units such as m; area is in square units such as m².

Why is the area of a triangle half base times height?

Two identical triangles fit together to make a parallelogram with area base times height, so one triangle is half of that.