Area & perimeter
Rectangles, triangles, trapezoids, and shapes made of them.
In your head, jot if needed · no calculator why?
Opens at level 17. You're level 1. You can read and practice here now.
the idea
Area is the space a flat shape covers, counted in squares: square metres, square feet. Perimeter is the distance around the edge, in plain metres or feet.
Rectangles are the base case: area is length × width. A triangle is half of a rectangle with the same base and height. A trapezoid, a four-sided shape with one pair of parallel sides, works like a rectangle whose width is the average of those two sides. Bigger shapes break into these pieces.
techniques
Rectangle: multiply or go around
- Area: length × width, in square units.
- Perimeter: add all four sides, which is 2 × (length + width).
worked example
A rectangle is 12 m by 7 m. What is its perimeter in metres?
- 2 × (12 + 7) = 2 × 19 = 38 m.
- Its area would be 12 × 7 = 84 square metres.
Answer: 38
Halve for triangles and trapezoids
- Triangle: base × height ÷ 2. The height is measured at a right angle to the base.
- Trapezoid: add the two parallel sides, halve, then multiply by the height.
worked example
A trapezoid has parallel sides of 6 cm and 10 cm and a height of 4 cm. What is its area in square centimetres?
- Average the parallel sides: (6 + 10) ÷ 2 = 8.
- Multiply by the height: 8 × 4 = 32 cm².
Answer: 32
Break it into rectangles
- A corner cut out: the whole rectangle minus the missing piece.
- Two rectangles joined: add their areas.
- Sketch it and label every length before you multiply.
worked example
A patio is a 9 m × 6 m rectangle with a 3 m × 2 m corner cut out. What is its area in square metres?
- Whole rectangle: 9 × 6 = 54.
- Missing corner: 3 × 2 = 6.
- 54 − 6 = 48 m².
Answer: 48
watch out for
- Giving the perimeter when the question asks for area, or the reverse. Square metres means area.
- Forgetting to halve for a triangle or a trapezoid.
- Adding the cut-out corner instead of subtracting it.
- Adding only two sides for a perimeter. A rectangle has four.
practice
Rectangles
worked example
A rectangle is 6 ft by 14 ft. What is its perimeter in feet?
Answer: 40
- Perimeter = 2 × (6 + 14) = 2 × 20 = 40 ft.
Triangles & trapezoids
worked example
A triangle has a base of 28 cm and a height of 7 cm. What is its area in square centimetres?
Answer: 98
- Area = ½ × base × height = ½ × 28 × 7 = 98 cm².
Composite shapes
worked example
An L-shaped garden is a 15 ft × 20 ft rectangle with a 3 ft × 6 ft corner cut out. What is its area in square feet?
Answer: 282
- Whole rectangle minus the missing corner: 15 × 20 − 3 × 6 = 300 − 18 = 282 ft².