POP

C5.2 Area and Perimeter

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01 — Explanation

What's the difference between perimeter and area?

Perimeter is the total distance around a shape's outside edge, in a linear unit (cm, m). Area is the flat space enclosed inside that boundary, in a squared unit ( $\text{cm}^2$ , $\text{m}^2$ ). Mixing the two up — adding lengths when area was asked for, or vice versa — is one of the most common errors at this level, so always check which one the question wants and give the matching unit.

02 — Explanation

What are the area formulas for rectangles, triangles, parallelograms and trapezia?

Shape	Area formula
Rectangle	$A = \text{length} \times \text{width}$
Triangle	$A = \frac{1}{2} \times \text{base} \times \text{perpendicular height}$
Parallelogram	$A = \text{base} \times \text{perpendicular height}$
Trapezium	$A = \frac{1}{2}(a+b) \times h$ , where $a, b$ are the parallel sides

Except for the triangle formula, none of these are printed on the exam paper — they must be memorised.

Common mistake

multiplying by a slanted side instead of the perpendicular height. Every one of these formulas needs the perpendicular (right-angle) distance to the base, never a slanted measurement — a diagram that shows a slanted side is often a deliberate hint to check which length is genuinely at right angles to the base.

03 — Explanation

Why does the parallelogram formula work, and how does the triangle formula relate to it?

A parallelogram can be cut along a diagonal into two identical triangles, so its area is exactly twice the area of one of those triangles — cutting a vertical slice off one end of a parallelogram and sliding it to the other side also turns it into a rectangle with the same base and height, which is why $A = \text{base}\times\text{height}$ works for both shapes. A triangle, being half of a parallelogram built on the same base and height, naturally picks up the extra $\frac{1}{2}$ in its formula.

04 — Explanation

How do I find the area and perimeter of a triangle when only three sides are known?

If the height isn't given directly, look for a right angle within the triangle (often signalled by two sides being given as the base and height already, or a right-angle marker in the diagram) before applying the area formula. The perimeter of any triangle is simply the sum of its three sides.

Worked example. A triangle has base $10\text{ cm}$ , perpendicular height $6\text{ cm}$ , and the other two sides measure $6.5\text{ cm}$ and $8\text{ cm}$ . Find its area and perimeter.

A = \frac{1}{2}\times10\times6 = 30\text{ cm}^2 \qquad P = 10+6.5+8 = 24.5\text{ cm}

05 — Explanation

How do I find the area of a trapezium?

Worked example. A trapezium has parallel sides of $9\text{ cm}$ and $15\text{ cm}$ , and perpendicular height $7\text{ cm}$ . Find its area.

9 + 15 = 24\text{ cm} \qquad A = \frac{1}{2}(24) \times 7 = 12 \times 7 = 84\text{ cm}^2

Exam tip

in the trapezium formula, $a$ and $b$ must be the two parallel sides — the two slanted (non-parallel) sides are never used in the area calculation, even though they may be labelled clearly in the diagram.