Perimeter, Area, and Volume

Measurement questions feature prominently on the CSSE 11+ maths paper. You need to be comfortable calculating the perimeter and area of 2D shapes, as well as the volume and surface area of 3D shapes. The CSSE often tests these skills through multi-step problems — for example, finding the area of a compound shape or working backwards from a given area to find a missing length.

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Perimeter

Perimeter is the total distance around the outside of a 2D shape, measured in units of length (cm, m, km, etc.).

Perimeter Formulas

Shape	Perimeter formula
Rectangle	2 × (length + width)
Square	4 × side
Triangle	side a + side b + side c
Circle (circumference)	pi × d or 2 × pi × r

Worked Example

A rectangle is 14 cm long and 9 cm wide. What is its perimeter?

• Perimeter = 2 × (14 + 9) = 2 × 23 = 46 cm

Compound Shape Perimeters

For compound shapes, find all the side lengths first (you may need to calculate missing lengths), then add them up.

CSSE Tip: Always label every side on the diagram before adding. This prevents you from missing a side.

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Area of 2D Shapes

Area is the space inside a flat shape, measured in square units (cm², m²).

Key Area Formulas

Shape	Formula
Rectangle	length × width
Square	side × side
Triangle	1/2 × base × perpendicular height
Parallelogram	base × perpendicular height
Trapezium	1/2 × (a + b) × h, where a and b are the parallel sides
Circle	pi × r²

Worked Examples

Rectangle: length = 15 cm, width = 8 cm

• Area = 15 × 8 = 120 cm²

Triangle: base = 12 cm, height = 7 cm

• Area = 1/2 × 12 × 7 = 42 cm²

Parallelogram: base = 10 cm, perpendicular height = 6 cm

• Area = 10 × 6 = 60 cm²

Trapezium: parallel sides = 8 cm and 12 cm, height = 5 cm

• Area = 1/2 × (8 + 12) × 5 = 1/2 × 20 × 5 = 50 cm²

Important: Always use the perpendicular (vertical) height, not the slant height!

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Area of Compound Shapes

A compound shape is made from two or more simple shapes joined together.

Method

1. Split the shape into rectangles, triangles, or other known shapes.
2. Calculate the area of each part.
3. Add the areas together (or subtract if there is a cut-out).

Worked Example

An L-shaped room has these measurements:

Part	Length	Width
Rectangle A	10 m	4 m
Rectangle B	6 m	3 m

• Area A = 10 × 4 = 40 m²
• Area B = 6 × 3 = 18 m²
• Total area = 40 + 18 = 58 m²

Worked Example: Finding a Missing Length

A rectangle has an area of 72 cm² and a width of 8 cm. What is its length?

• Area = length × width
• 72 = length × 8
• length = 72 ÷ 8 = 9 cm

What is its perimeter?

• Perimeter = 2 × (9 + 8) = 2 × 17 = 34 cm

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Circles

Measurement	Formula
Circumference (perimeter)	pi × diameter or 2 × pi × radius
Area	pi × radius²

Use pi = 3.14 or pi = 22/7 as instructed.

Find the area of a circle with radius 7 cm (use pi = 22/7):

• Area = 22/7 × 7 × 7 = 22/7 × 49 = 154 cm²

Find the circumference of a circle with diameter 10 cm (use pi = 3.14):