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Build: Area

Measurement · step 3 of 5

A question to start

You have 20 metres of fence to make a rectangular pen for a rabbit. You could make it long and thin, 9 m by 1 m. Or 6 m by 4 m. Or a square, 5 m by 5 m. All three use exactly 20 m of fence. Does the rabbit get the same amount of grass in each?

No. The thin pen holds 9 square metres of grass, the 6 by 4 pen holds 24, and the square holds 25. The fence round the edge (the perimeter) is the same, but the space inside is very different. That space is the area, and this page shows how to find it for every common shape.

The idea

What area means

Area is the amount of flat surface a shape covers. We measure it by counting how many unit squares fit inside, with no gaps and no overlaps.

  • A square centimetre (cm²) is a square 1 cm on each side, about the size of a fingernail.
  • A square metre (m²) is a square 1 m on each side. A single bed covers about 2 m².
  • A square kilometre (km²) is a square 1 km on each side, used for cities and countries.

The little 2 means "squared": it reminds you that area has two directions, length and width.

Rectangles and squares

A rectangle 6 cm long and 4 cm wide holds 4 rows of 6 unit squares: \(4 \times 6 = 24\) cm². In general:

\[\text{area of a rectangle} = \text{length} \times \text{width}\]

A square with side \(s\) has area \(s \times s = s^2\). That is why "squared" means "times itself".

Parallelograms

A parallelogram is a slanted rectangle: four sides, with opposite sides parallel. Cut off the triangle at one end and slide it to the other end. It fits exactly, and the shape becomes a rectangle.

base height → same base, same height

The rectangle has the same base and the same height as the parallelogram, so

\[\text{area of a parallelogram} = \text{base} \times \text{height}\]

The height is the straight-up distance between the base and the side opposite it, measured at a right angle to the base. It is not the length of the slanted side.

Triangles

Any triangle is exactly half of a parallelogram: put two copies of the triangle together, one turned round, and they make a parallelogram with the same base and height. So

\[\text{area of a triangle} = \tfrac12 \times \text{base} \times \text{height}\]

Again the height is measured straight up from the base to the opposite corner. For a right triangle the two short sides are the base and the height.

Trapezoids

A trapezoid has one pair of parallel sides, of lengths \(a\) and \(b\), a distance \(h\) apart. Two copies, one turned upside down, fit together into a parallelogram with base \(a + b\) and height \(h\). The trapezoid is half of it:

\[\text{area of a trapezoid} = \tfrac12 (a + b) \times h\]

In words: average the two parallel sides, then multiply by the height.

Circles

As the Geometry ladder shows by cutting a circle into thin wedges, a circle of radius \(r\) has area

\[A = \pi r^2 \qquad (\pi \approx 3.14)\]

Shapes made of shapes

Most real shapes are combinations. Find the area of an odd shape by cutting it into simple pieces and adding, or by taking a whole simple shape and subtracting the part that is missing. Often both work; pick the easier one.

Converting area units

Be careful here. 1 m is 100 cm, but 1 m² is not 100 cm². A square metre is 100 cm long and 100 cm wide, so it holds \(100 \times 100 = 10\,000\) square centimetres.

Length Area
metres and centimetres 1 m = 100 cm 1 m² = 10 000 cm²
kilometres and metres 1 km = 1000 m 1 km² = 1 000 000 m²

Farmland is often measured in hectares: 1 hectare is a square 100 m on each side, 10 000 m². A football field is a bit less than one hectare.

Worked examples

1. The rabbit pens. A 9 m by 1 m pen has area \(9 \times 1 = 9\) m². A 6 m by 4 m pen has \(6 \times 4 = 24\) m². A 5 m by 5 m pen has \(5^2 = 25\) m². For a fixed perimeter, the square gives the most area of any rectangle.

2. An L-shaped room. A room is a 10 m by 8 m rectangle with a 4 m by 3 m corner missing (the same shape as the garden on the Explore page). Its area?

  1. Subtract: whole rectangle \(10 \times 8 = 80\) m², minus the corner \(4 \times 3 = 12\) m², gives 68 m².
  2. Check by adding instead: cut it into a 6 m by 8 m piece (\(48\) m²) and a 4 m by 5 m piece (\(20\) m²). \(48 + 20 = 68\) m² ✓.

3. A sail. A triangular sail has a base of 3 m and a height of 5 m. Area: \(\tfrac12 \times 3 \times 5 = 7.5\) m².

4. A ring. A flat washer is a circle of radius 5 mm with a hole of radius 3 mm. Its area is the big circle minus the hole: \(\pi \times 5^2 - \pi \times 3^2 = 25\pi - 9\pi = 16\pi \approx 50.3\) mm².

Try it

  1. Find the area of a rectangle 8 m long and 5.5 m wide.
  2. A triangle has base 10 cm and height 7 cm. Find its area.
  3. A parallelogram has base 12 cm, slanted sides of 5 cm, and height 4 cm. Find its area.
  4. A trapezoid has parallel sides of 6 cm and 10 cm, 5 cm apart. Find its area.
  5. Find the area of a circle of radius 3 m, to one decimal place.
  6. A poster has an area of 3.2 m². What is that in cm²?
Answers
  1. \(8 \times 5.5 = 44\) m².
  2. \(\tfrac12 \times 10 \times 7 = 35\) cm².
  3. \(12 \times 4 = 48\) cm². The 5 cm slanted side is not used: the height is what counts.
  4. \(\tfrac12 \times (6 + 10) \times 5 = 8 \times 5 = 40\) cm².
  5. \(\pi \times 3^2 = 9\pi \approx 28.3\) m².
  6. Each m² is 10 000 cm², so \(3.2 \times 10\,000 = 32\,000\) cm².

Common mistake

Using the slanted side as the height. For parallelograms, triangles and trapezoids, the height must be measured at a right angle to the base. The slanted side is always longer than the height, so using it gives an area that is too big. If a diagram shows a dashed line meeting the base with a small square corner, that dashed line is the height.

Where it leads