Area of Rectangles and Triangles: KS3 Maths Guide with Worked Examples

The area of a shape is the amount of two-dimensional space it covers, measured in square units such as cm² or m². For a rectangle, the area is length × width. For a triangle, it is half the base × the perpendicular height. These two formulae are the building blocks for finding the area of almost every 2D shape at KS3.

Area of a rectangle

Formula: A = l × w (also written A = lw)

Where l is the length and w is the width.

Why does this formula work?

Imagine a rectangle that is 4 cm wide and 3 cm tall. Divide it into 1 cm squares: you get 4 squares along each row and 3 rows, giving 4 × 3 = 12 squares. Each square has area 1 cm², so the total area is 12 cm².

Worked example 1 — rectangle

A rectangle has length 9 cm and width 5 cm. Find its area.

A = l × w = 9 × 5 = 45

Answer: 45 cm²

Worked example 2 — rectangle with decimal dimensions

A rectangular garden is 7.5 m long and 4 m wide. Find its area.

A = 7.5 × 4 = 30

Answer: 30 m²

Worked example 3 — finding a missing length from the area

A rectangle has area 56 cm² and width 7 cm. Find the length.

A = l × w, so 56 = l × 7. Divide both sides by 7: l = 56 ÷ 7 = 8.

Answer: length = 8 cm

Area of a triangle

Formula: A = ½ × b × h (also written A = bh/2)

Where b is the base and h is the perpendicular height (the height measured at right angles to the base, not along a slanted side).

Why is the formula half of base × height?

Any triangle can be enclosed in a rectangle with the same base and height. The triangle always takes up exactly half the area of that rectangle. That is why the formula has a ½.

For a right-angled triangle, this is obvious — the diagonal cuts the rectangle in half. For non-right-angled triangles, if you draw two right-angled triangles either side of the height, each one is half of a smaller rectangle, and they combine to give half the total rectangle.

Worked example 4 — right-angled triangle

A right-angled triangle has base 8 cm and height 5 cm. Find its area.

A = ½ × 8 × 5 = ½ × 40 = 20

Answer: 20 cm²

Worked example 5 — non-right-angled triangle

A triangle has base 12 cm and perpendicular height 7 cm. Find its area.

A = ½ × 12 × 7 = ½ × 84 = 42

Answer: 42 cm²

Worked example 6 — finding the perpendicular height from area

A triangle has area 30 cm² and base 10 cm. Find the perpendicular height.

30 = ½ × 10 × h, so 30 = 5h, giving h = 30 ÷ 5 = 6.

Answer: h = 6 cm

The perpendicular height — a common source of error

The perpendicular height is always measured at right angles to the base. It is NOT the slanted side of the triangle.

Correct use of height:

Suppose a triangle has a slant side of 10 cm and a base of 8 cm, but the perpendicular height (drawn from the opposite vertex straight down to the base) is 6 cm. The area is:

A = ½ × 8 × 6 = 24 cm²

Using the slant side of 10 cm would give ½ × 8 × 10 = 40 cm² — wrong.

Always identify which measurement is perpendicular before substituting into the formula.

Compound shapes using rectangles and triangles

Many KS3 problems ask for the area of a shape that can be divided into rectangles and triangles (or a rectangle with a triangle cut away).

Worked example 7 — L-shape (two rectangles)

An L-shaped room has an overall width of 10 m and an overall height of 8 m. A rectangular section 4 m × 3 m has been cut from the top-right corner.

Total rectangle area: 10 × 8 = 80 m²

Cut-away rectangle: 4 × 3 = 12 m²

L-shape area: 80 − 12 = 68 m²

Answer: 68 m²

Worked example 8 — rectangle plus triangle (pentagon)

A shape consists of a rectangle 6 cm × 4 cm with a triangle on top. The triangle has the same base (6 cm) as the rectangle and a perpendicular height of 3 cm.

Rectangle: 6 × 4 = 24 cm²

Triangle: ½ × 6 × 3 = 9 cm²

Total: 24 + 9 = 33 cm²

Answer: 33 cm²

Units of area

Area is always measured in square units:

If dimensions are in cm, area is in cm²

If dimensions are in m, area is in m²

If dimensions are in mm, area is in mm²

Converting between units of area:

1 m² = 10 000 cm² (because 1 m = 100 cm, and 100² = 10 000)

1 cm² = 100 mm² (because 1 cm = 10 mm, and 10² = 100)

This conversion is a common trap: students multiply by 100 instead of 10 000 when converting m² to cm².

Summary of formulae

Shape | Formula | Variables
Rectangle | A = lw | l = length, w = width
Triangle | A = ½bh | b = base, h = perpendicular height
Square | A = s² | s = side length

How area fits the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study requires pupils to "derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms." Rectangle and triangle area are core knowledge that underpins every other area formula taught at KS3. For example, the area of a parallelogram (A = bh) uses the same base-times-perpendicular-height principle as the triangle formula, as set out in the statutory national curriculum for Key Stage 3 and 4.

Common mistakes

Mistake 1 — Using the slant height instead of the perpendicular height. Always check that the height is drawn at 90° to the base. In exam diagrams, a right-angle marker shows the perpendicular.

Mistake 2 — Forgetting the ½ in the triangle formula. ½ × 6 × 4 = 12, not 6 × 4 = 24. Forgetting the ½ doubles the answer.

Mistake 3 — Missing the square on units. Area = 30 cm², not 30 cm. Omitting the superscript 2 costs a mark.

Mistake 4 — Adding dimensions from different shapes in a compound problem. In a compound shape, calculate each section separately and then add (or subtract) the areas.

Frequently asked questions

What is the difference between area and perimeter?

Area measures the space inside a shape (in square units). Perimeter measures the total distance around the outside of a shape (in linear units — cm, m, etc.). For a 6 cm × 4 cm rectangle, area = 24 cm² and perimeter = 2(6 + 4) = 20 cm. These are different quantities with different units.

Can I use any side of a triangle as the base?

Yes. You can choose any side as the base, as long as you pair it with the perpendicular height measured to that base. Different choices of base will give different-looking calculations but must produce the same area. For example, rotating the triangle just relabels the base and corresponding height.

Why does the perpendicular height matter and not the slant side?

The formula A = ½bh is derived from the area of a rectangle. The rectangle's height is always perpendicular to its base. Only the perpendicular height tells you how far the triangle extends in the direction at right angles to the base — which is what determines area. The slant side is longer than the perpendicular height, so using it would overestimate the area.

How do I find the area of a triangle when I only know three side lengths?

At KS3 you are not expected to know this without a perpendicular height given in the problem. At GCSE you learn the formula A = ½ab sin C, which uses two sides and the included angle. For now, if the perpendicular height is not marked, assume you need to calculate or be given it before you can find the area.

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