Perimeter & Area of Polygons
Rectangles, triangles, parallelograms, and trapezoids. Learn which formula goes with which shape — and why the height is never the slant side.
1 Perimeter vs. area
Two different measurements. Mixing them up is the most common error in this skill.
Perimeter — the fence
The distance around a shape. Add up all the side lengths. Units: plain units (cm, m, ft).
P = sum of all sides
Area — the carpet
The amount of surface inside a shape. Units: square units (cm², m², ft²).
Fencing a garden → perimeter. Sodding (covering) it → area.
2 The four formulas
Every polygon area formula below is built from base × height — with the triangle and trapezoid taking a fraction of it.
Rectangle
A = l × w
P = 2l + 2w
Triangle
A = ½ × b × h
A triangle is half its bounding rectangle — never forget the ½.
Parallelogram
A = b × h
The height (5, blue dashed) is perpendicular to the base — never the slant side (6, red).
Trapezoid
A = ½ × (b₁ + b₂) × h
Average the two bases first, then multiply by the height.
3 Worked examples
Missing sides and composite figures — the two skills that turn formulas into problem solving.
Example 1 — working backwards
- A rectangle has area 48 cm² and length 8 cm. Find the width.
- A = l × w → 48 = 8 × w.
- Divide: w = 48 ÷ 8 = 6 cm.
Example 2 — composite figure
- Split the figure: a 10×6 rectangle plus a triangle with base 10 and height 4.
- Rectangle: 10 × 6 = 60. Triangle: ½ × 10 × 4 = 20.
- Total area: 60 + 20 = 80.
Example 3 — perimeter
- A 9 m by 4 m garden needs fencing. Find the perimeter.
- P = 2l + 2w = 2(9) + 2(4) = 18 + 8 = 26 m.
- Units are plain meters — perimeter is a length, not an area.