Key takeaways
- Area is the flat space inside a shape, always measured in square units such as m².
- Rectangle: length × width. Triangle: ½ × base × height. Circle: π × radius².
- A triangle's height must be at a right angle to its base, not along a slanted side.
- Split any odd shape into rectangles and triangles, then add the areas.
In this guide
What is area?
Area is the amount of flat surface inside a shape. Imagine covering a floor with square tiles that are 1 m on each side. The number of tiles you need is the area in square metres (m²).
That's why area always has a small ² on its unit. You are counting squares, not lengths.
Why does area matter?
Area is the number shops use to sell you things. Flooring, paint, grass seed, fabric, solar panels and land are all priced by area. Get the area right and you buy the right amount the first time.
Area also grows faster than you expect. A square room that is 3 m wide has an area of 9 m². Make it 6 m wide and the area is 36 m², four times as much, not twice.
The formulas
Most area formulas grow from one simple idea:
| Shape | Formula | What the letters mean |
|---|---|---|
| Square | A = s² | s = side length |
| Rectangle | A = l × w | l = length, w = width |
| Triangle | A = ½ × b × h | b = base, h = height |
| Parallelogram | A = b × h | b = base, h = height |
| Trapezoid | A = ½ × (a + b) × h | a, b = the two parallel sides |
| Circle | A = π × r² | r = radius, π ≈ 3.14159 |
See the pattern? A parallelogram is a rectangle with one end slid over, so it keeps b × h. A triangle is half a parallelogram, so it gets the ½. A trapezoid uses the average of its two parallel sides as its width.
How to calculate area step by step
- 1Name the shape. Is it a rectangle, triangle, circle or trapezoid? If it is none of these, split it up.
- 2Measure what the formula needs. For a triangle, that is the base and the perpendicular height.
- 3Make the units match. Change everything to metres, or everything to centimetres.
- 4Substitute and calculate. Replace the letters with your numbers.
- 5Write square units. Your answer is in m², cm² or ft².
When you know all three sides of a triangle
Sometimes you know the three sides but not the height. Then use Heron's formula. First find half the perimeter, s = (a + b + c) ÷ 2. Then the area is √(s(s − a)(s − b)(s − c)). For sides 5, 6 and 7, s = 9 and the area is √(9 × 4 × 3 × 2) = √216 ≈ 14.70. The Heron's formula calculator does this for you, step by step.
Common mistakes to avoid
- Forgetting the ½. Triangles and trapezoids both need it.
- Using the diameter in the circle formula. Halve it first. A 8 cm wide circle has a radius of 4 cm, so its area is
π × 4² ≈ 50.27 cm². - Mixing m and cm. 1 m² is 10,000 cm², not 100 cm², because each side is 100 cm.
- Squaring the wrong thing. In
πr², only r is squared, not π.
Tips and tricks
- Subtract to find a border. The area of a path around a pool is the big rectangle minus the pool.
- Check with an estimate. A circle's area is a little more than 3 × r². If you get something much bigger, you probably used the diameter.
- Use the matching tool. Try the trapezoid area calculator or the triangle area calculator to confirm your working.
Where you'll use it in real life
- Flooring and tiles: a 4 m × 3.5 m room needs
4 × 3.5 = 14 m²of flooring, plus a little extra for cutting. - Painting: paint tins say how many m² they cover.
- Gardening: grass seed and fertiliser are measured per m².
- Land and property: plots and flats are bought and sold by area.
Quick summary
Area counts the square units inside a shape. Pick the formula for your shape, measure the right lengths (perpendicular height, radius not diameter), keep the units the same, and finish with square units. Split odd shapes into simple pieces, and use the free calculators linked above to check each step.
Frequently asked questions
How do I calculate the area of an irregular shape?
Break the shape into rectangles and triangles, find the area of each piece, then add them together. For cut-out parts, subtract their area instead.
What units should I use for area?
Area is always in square units such as cm², m² or ft². Make sure every length uses the same unit before you calculate.
Why is the triangle area formula half of base times height?
Two identical triangles fit together to make a parallelogram with area base × height. One triangle is half of that.
How many cm² are in 1 m²?
There are 10,000 cm² in 1 m², because a 1 m square is 100 cm by 100 cm, and 100 × 100 = 10,000.
