Area & Perimeter

What are Area and Perimeter?

Perimeter is the total distance around the outside of a shape. It is a length, so it is measured in units such as cm, m, or km.

Area is the amount of space enclosed inside a shape. It is measured in square units such as cm², m², or km².

These two concepts are often confused. A key way to remember them: perimeter is like a fence around a field (one-dimensional distance), while area is like the grass inside the field (two-dimensional space).

Perimeter

To find the perimeter of any shape, add up all the side lengths.

Rectangle: P = 2(l + w) where l = length and w = width

• Rectangle 8cm × 5cm: P = 2(8 + 5) = 2 × 13 = 26cm

Square: P = 4s where s = side length

• Square with side 7cm: P = 4 × 7 = 28cm

Triangle: P = a + b + c (sum of all three sides)

• Sides 5cm, 7cm, 9cm: P = 5 + 7 + 9 = 21cm

For irregular polygons, measure and add every side individually.

Area of Rectangles and Squares

Rectangle: A = l × w

• Length 8cm, width 5cm: A = 8 × 5 = 40cm²

Square: A = s²

• Side 9cm: A = 9² = 81cm²

Note that the area of a rectangle (40cm²) is different from its perimeter (26cm) — always check which one the question is asking for.

Area of Triangles and Parallelograms

Triangle: A = 1/2 × base × perpendicular height

• Base 10cm, height 6cm: A = 1/2 × 10 × 6 = 30cm²

The height must be perpendicular (at 90°) to the base — not a slanted side.

Parallelogram: A = base × perpendicular height

• Base 12cm, height 5cm: A = 12 × 5 = 60cm²

A parallelogram has the same area formula as a rectangle when the perpendicular height is used.

Area of Trapezium

A trapezium has one pair of parallel sides (called a and b) and a perpendicular height h.

Trapezium: A = 1/2 × (a + b) × h

Worked example: Parallel sides 5cm and 9cm, height 4cm

• A = 1/2 × (5 + 9) × 4 = 1/2 × 14 × 4 = 28cm²

Think of it as finding the average of the two parallel sides, then multiplying by the height.

Circles: Circumference and Area

For circles, all formulas involve π (pi) ≈ 3.14159...

Circumference (perimeter of a circle): C = πd = 2πr

• Radius 7cm: C = 2 × π × 7 ≈ 2 × 3.14 × 7 = 43.96cm

Area of a circle: A = πr²

• Radius 7cm: A = π × 7² = π × 49 ≈ 3.14 × 49 = 153.86cm²

Remember: circumference uses radius or diameter; area always uses radius squared.

Key Terms

Term	Meaning
Perimeter	Total length around the outside of a shape
Area	Space enclosed inside a shape
Perpendicular height	Height measured at 90° to the base
Radius (r)	Distance from centre to edge of a circle
Diameter (d)	Distance across a circle through the centre (d = 2r)
π (pi)	Approximately 3.14159, the ratio of circumference to diameter

Composite Shapes

A composite shape is made by joining two or more simple shapes together. To find the area:

1. Split the composite shape into rectangles, triangles, and/or semicircles.
2. Calculate the area of each part.
3. Add (or subtract) the parts.

Example: An L-shaped room can be split into two rectangles. Find each rectangle's area, then add them together.

If a shape has a region removed (e.g. a square with a circular hole), calculate the area of the whole shape then subtract the area of the removed part.

Units and Conversions

Always use consistent units throughout a calculation.

Conversion	Equivalent
1 m = 100 cm	1 m² = 10,000 cm²
1 km = 1,000 m	1 km² = 1,000,000 m²

Note that unit conversions for area require squaring the linear conversion factor — a common source of error.

Common Mistakes

• Confusing area and perimeter — re-read the question carefully and check what unit the answer should be in.
• Using slant height instead of perpendicular height — always use the height that is at 90° to the base.
• Forgetting to square r in the circle area formula — A = πr², not πr.
• Using diameter instead of radius — check which measurement you have been given before applying a formula.
• Not converting units — mixing cm and m in the same calculation gives a wrong answer.

Tips and Tricks

• Memorise the key formulas as a group: rectangle (l × w), triangle (1/2 × b × h), circle area (πr²), circumference (2πr).
• For composite shapes, draw dotted lines to show how you are splitting them — this prevents double-counting.
• A quick check: area answers must always have squared units (cm², m²); perimeter answers have plain units (cm, m).
• If you know the perimeter of a square, divide by 4 to get the side, then square for the area — a two-step shortcut.