Area, Perimeter & Volume

Measuring shapes is a core part of the GL 11+ Mathematics exam. You need to know how to calculate the perimeter (distance around), area (space inside), and volume (space inside a 3D shape). This lesson gives you the formulas and methods for every shape you might encounter.

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Perimeter

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

How to Find the Perimeter

Add up the lengths of all the sides.

Worked Example: A rectangle is 8 cm long and 5 cm wide. Find its perimeter.

• Perimeter = 8 + 5 + 8 + 5 = 26 cm
• Or use the formula: Perimeter = 2 × (length + width) = 2 × (8 + 5) = 2 × 13 = 26 cm

Perimeters of Compound Shapes

For irregular shapes, find any missing side lengths first, then add all sides.

Worked Example:

    6 cm
   ________
  |        |
  |        | 3 cm
  |   _____|
  |  | 2 cm
  |  |
  |__|
   4 cm

This L-shaped figure has sides: 6, 3, 2, (3 - unknown), 4, (6 - 2).

• Missing vertical side: 4 - 3 = ... work out all sides from the diagram.
• Add them all together for the perimeter.

Exam Tip: Label every side on the diagram before you start adding. This helps avoid missing a side.

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Area of 2D Shapes

Area is the amount of space inside a flat shape. It is measured in square units (cm², m², km²).

Key Area Formulas

Shape	Formula	Diagram hint
Rectangle	length × width	l × w
Square	side × side = side²	s²
Triangle	½ × base × height	½ × b × h
Parallelogram	base × perpendicular height	b × h
Trapezium	½ × (a + b) × h	a and b are the parallel sides

Worked Examples

Rectangle: length = 12 cm, width = 7 cm

• Area = 12 × 7 = 84 cm²

Triangle: base = 10 cm, height = 6 cm

• Area = ½ × 10 × 6 = 30 cm²

Parallelogram: base = 9 cm, perpendicular height = 4 cm

• Area = 9 × 4 = 36 cm²

Important: For triangles and parallelograms, always use the perpendicular (vertical) height — not the slant height!

Trapezium: parallel sides = 5 cm and 9 cm, height = 4 cm

• Area = ½ × (5 + 9) × 4 = ½ × 14 × 4 = 28 cm²

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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 cutout).

Worked Example: An L-shape

   8 cm
   __________
  |          |
  |          | 3 cm
  |     _____|
  |    |
  |    | 5 cm
  |____|
   4 cm

Split into two rectangles:

• Rectangle 1: 4 × (3 + 5) = 4 × 8 = 32 cm²
• Rectangle 2: (8 - 4) × 3 = 4 × 3 = 12 cm²
• Total area = 32 + 12 = 44 cm²

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Circles (Bonus — Sometimes Tested)

Measurement	Formula
Circumference	π × d or 2πr
Area	π × r²

Where π ≈ 3.14 (or use 22/7), d = diameter, r = radius.