Area and Perimeter of Shapes

Learn how to calculate the area and perimeter of rectangles, triangles, circles, trapeziums and compound shapes — with real-world examples like carpeting a room.

Perimeter is the distance around the outside of a shape — like walking around the edge of a field.Areais the space inside the shape — like how much carpet you'd need to cover a floor.

Rectangle

Area = length × width | Perimeter = 2(length + width)

Worked Example 1 — Carpeting a Room

A room is 5m long and 4m wide. Carpet costs £12 per m².

Area = 5 × 4 = 20 m²
Cost = 20 × £12 = £240
Perimeter (for skirting board) = 2(5 + 4) = 18 m

Triangle

Area = ½ × base × height

The height must be perpendicular (at right angles) to the base — not the slanted side.

Circle

Area = π × r² | Circumference = 2 × π × r

Where r is the radius (half the diameter) and π ≈ 3.14159 (use the π button on your calculator).

Worked Example 2 — Circular Garden

A circular pond has a diameter of 6m. Find its area and the length of fencing needed.

Radius = 6 ÷ 2 = 3m
Area = π × 3² = π × 9 = 28.27 m²
Circumference (fencing) = 2 × π × 3 = 18.85 m

Trapezium

Area = ½ × (a + b) × h

Where a and b are the two parallel sides and h is the height between them.

Compound Shapes

Break complex shapes into simpler ones (rectangles + triangles), calculate each area, then add them up.

Worked Example 3 — L-Shaped Room

An L-shaped room: the main part is 6m × 4m, and the extension is 3m × 2m.

Main rectangle: 6 × 4 = 24 m²
Extension: 3 × 2 = 6 m²
Total area = 24 + 6 = 30 m²

Units

Measurement	Unit	Conversion
Perimeter	cm, m, km	100 cm = 1 m
Area	cm², m²	10,000 cm² = 1 m²
Area (large)	hectares	10,000 m² = 1 hectare

Try It Yourself

Find the area of a triangle with base 12 cm and height 8 cm. (Answer: 48 cm²)
A circle has radius 5 m. Find its area. (Answer: 78.54 m²)
A garden is 10m × 8m. How much fencing for the perimeter? (Answer: 36 m)