Perimeter and Area of 2D Shapes — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the distance around a flat shape and the space inside it. By the end of this guide you should be able to find the perimeter and area of rectangles, triangles, parallelograms and trapeziums, and handle compound shapes made by joining or cutting away simple ones.

You should also be able to work backwards from an area to a missing length, deal with parts of circles, convert between units of area, and recognise which height a formula actually wants.

The organising idea is that every area formula is a rectangle in disguise. A parallelogram is a rectangle with a triangle moved from one end to the other, which is why its area is base × height. A triangle is half a parallelogram, which is where the half comes from. A trapezium is a rectangle whose width is the average of its two parallel sides. Seeing the formulas this way means you can rebuild any of them if memory fails, and it makes clear why the perpendicular height is always the one required — a slanted side would not be the side of the rectangle.

Key terms and definitions

Perimeter — the total distance around the outside of a shape, measured in ordinary length units such as cm.

Area — the amount of surface a shape covers, measured in square units such as cm².

Perpendicular height — the height measured at right angles to the base. This is what every area formula means by height.

Parallelogram — a quadrilateral with two pairs of parallel sides.

Trapezium — a quadrilateral with exactly one pair of parallel sides.

Compound shape — a shape made by joining or removing simple shapes.

Core concepts

Perimeter

Add every side length that lies on the outside of the shape.

Two things go wrong. First, the units must match before adding, so a shape with sides in centimetres and metres needs converting first. Second, on a compound shape, some edges are internal joins and are not part of the perimeter — only the outside boundary counts.

Where a shape has a curved edge, the perimeter includes the arc length, and any straight edges must be added to it. The perimeter of a semicircle is half the circumference plus the diameter across the flat side; leaving the diameter out is the standard error.

The four area formulas

Rectangle = length × width. This is the one all the others are built from.

Parallelogram = base × height. Cutting a triangle from one end and sliding it to the other turns a parallelogram into a rectangle of the same base and height, so the areas match. Note there is no halving, which is a frequent slip.

Triangle = ½ × base × height. Two identical triangles fit together to make a parallelogram, so a triangle is half of one.

Trapezium = ½ × (a + b) × h, where a and b are the parallel sides. Adding the parallel sides and halving gives their average, and the formula is then that average multiplied by the height — a rectangle of average width.

Which height?

Every one of these formulas needs the perpendicular height, measured at right angles to the base, not the length of a slanted side.

A question will often give both, precisely to see whether you can tell them apart. In a triangle with sides of 5 cm and 13 cm and a perpendicular height of 4 cm, it is the 4 that goes into the formula.

The reason goes back to the rectangle. The height is the distance between the base and the opposite side, which is what the rectangle's width would be; a slanted edge is longer than that distance.

Compound shapes

Split the shape into simple pieces, work out each area, then add. Alternatively, find the area of a larger shape and subtract the piece cut away.

Both routes work, and choosing the shorter one is worth a moment's thought. A rectangle with a square removed from a corner is much quicker by subtraction than by splitting into three rectangles.

Label each piece and write its area separately. A single unexplained number earns far fewer marks than a visible breakdown, and it also makes an arithmetic slip easy to find.

Any missing side lengths are found by comparing opposite sides: in a rectangular compound shape, the lengths along the top must add to the same total as those along the bottom.

Working backwards

Questions often give an area and ask for a missing length, which turns the formula into an equation.

A triangle of area 24 cm² with a base of 8 cm: substitute into ½ × 8 × h = 24, so 4h = 24 and h = 6 cm.

A rectangle of area 45 cm² with a length of 9 cm has a width of 45 ÷ 9 = 5 cm.

Units

Area is always in square units, and forgetting the square loses marks even when the number is right.

Converting between area units needs care, because the conversion factor is squared. There are 100 cm in a metre, so there are 100 × 100 = 10 000 cm² in a square metre — not 100.

The same applies to 1 m² = 10 000 cm² and 1 cm² = 100 mm².

Parts of circles

A circle has area πr² and circumference πd, which is the same as 2πr.

For a semicircle, halve both. For a quarter circle, take a quarter.

Remember the distinction between radius and diameter: the area formula uses the radius, so a circle of diameter 14 cm has radius 7 cm and area 49π cm².

When a compound shape includes a semicircular end, its area is added to the rest, but its straight diameter is an internal join and does not appear in the perimeter.

Worked examples

Example 1: A trapezium

Find the area of a trapezium whose parallel sides are 7 cm and 11 cm, with a perpendicular height of 6 cm.

Add the parallel sides: 7 + 11 = 18.

Halve that to get the average width: 9.

Multiply by the height: 9 × 6 = 54 cm².

The check is that the answer lies between the areas of two rectangles — one 7 by 6 (42 cm²) and one 11 by 6 (66 cm²) — which it does.

Example 2: A compound shape by subtraction

A rectangle measures 10 cm by 8 cm. A square of side 3 cm is cut from one corner. Find the area remaining and the perimeter of the resulting shape.

The rectangle's area is 10 × 8 = 80 cm². The square's area is 3 × 3 = 9 cm².

The remaining area is 80 − 9 = 71 cm².

The perimeter is more subtle. Removing a square from a corner replaces two edge pieces of 3 cm each with two new edges of 3 cm each, so the total distance around is unchanged: 2 × (10 + 8) = 36 cm.

That result surprises people, and it is worth tracing the boundary carefully rather than assuming the perimeter must fall.

Example 3: Working backwards

A parallelogram has an area of 72 cm² and a base of 12 cm. Find its perpendicular height.

The formula is base × height, with no halving.

So 12 × h = 72, giving h = 6 cm.

Had the shape been a triangle, the formula would have carried a half, giving ½ × 12 × h = 72 and h = 12 cm — twice as much. Identifying the shape before substituting is what decides this.

Common mistakes and how to avoid them

Halving a parallelogram. Its area is base × height with no half. The half belongs to the triangle.

Using a slanted side as the height. Every formula wants the perpendicular height.

Including internal edges in a perimeter. Only the outside boundary counts. Trace around the shape with a finger.

Forgetting the diameter on a semicircle's perimeter. Half the circumference plus the flat edge.

Confusing radius with diameter. The formulas use the radius, which is half the diameter.

Converting area units with the wrong factor. 1 m² is 10 000 cm², because the conversion is squared.

Dropping the square on the units. Area is cm², m² or mm², never plain cm.

Exam technique for "Perimeter and Area"

Write the formula down before substituting into it. That line earns a method mark even when the arithmetic afterwards goes wrong.

For compound shapes, draw the split on the diagram, label the pieces, and give each piece's area on its own line before adding or subtracting.

Check which height the question has given. Where two lengths are offered, the perpendicular one is the one to use, and the other is a deliberate distractor.

Decide whether adding or subtracting is quicker before starting a compound shape. Both are valid, but one is usually half the work.

State units on every answer, with the square for areas.

When working backwards from an area, write the formula as an equation and solve it rather than trying to spot the answer.

Quick revision summary

Every area formula is a rectangle in disguise. Rectangle = length × width. Parallelogram = base × height (no halving). Triangle = ½ × base × height, because it is half a parallelogram. Trapezium = ½ × (a + b) × h, the average of the parallel sides times the height.

Every formula needs the perpendicular height, never a slanted side.

Perimeter is the distance around the outside only — internal joins do not count, and a semicircular edge needs its diameter added to the half-circumference.

For compound shapes, split and add, or take a larger shape and subtract the cut-out. Label each piece and show its area separately.

To find a missing length, substitute into the formula and solve the resulting equation.

A circle has area πr² and circumference πd; halve both for a semicircle. The formulas use the radius.

Area units are squared, and so are their conversions: 1 m² = 10 000 cm².