Maths
How do you find the area of a triangle?
Quick answer
The area of a triangle is half of its base times its perpendicular height: A = ½ × b × h. A triangle with base 8 cm and height 5 cm has an area of 20 square centimetres.
In one sentence
The area of a triangle is half of its base multiplied by its perpendicular height.
Explained
Area measures how much flat space a shape covers. For a rectangle, the area is length × width. A triangle is exactly half of a rectangle or parallelogram, which gives a simple formula:
Area = ½ × base × height
Here is why. Draw a triangle, then turn a copy upside down and fit it against the first. Together they form a parallelogram, with area base × height. Each triangle is half of that.
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The base can be any side you choose. The height is the straight, perpendicular distance from the base up to the opposite corner. It makes a right angle with the base. It is not the slanted side. For an obtuse triangle, the height line may fall outside the triangle, but the formula still works.
Example: a triangle has base 8 cm and height 5 cm.
Step by step
- Multiply base by height.8 × 5 = 40.
- Halve it.40 ÷ 2 = 20.
- Add the units.The area is 20 cm², said "20 square centimetres".
You can also run the formula backwards. If the area is 24 cm² and the base is 6 cm, then ½ × 6 × h = 24, so 3 × h = 24 and h = 8 cm.
For a right-angled triangle, the two sides at the right angle can be used as base and height. Compare with the general idea of area, and review the types of triangles. The shape also links to perimeter, which is the distance around.
Examples
| Base | Height | Area |
|---|---|---|
| 8 cm | 5 cm | 20 cm² |
| 10 cm | 6 cm | 30 cm² |
| 7 m | 4 m | 14 m² |
| 3 cm | 4 cm (right angle) | 6 cm² |
Common mistakes
- Using the slanted side as height. Use the perpendicular height.
- Forgetting to halve. Base × height is for a rectangle.
- Wrong units. Area is in square units, such as cm², not cm.
Quick quiz
Area of a triangle
Check what you know with 4 quick questions.
Question 1 of 4
What is the area of a triangle with base 10 cm and height 6 cm?
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Related topics
Area
Area is the amount of flat space inside a shape, measured in square units like cm² or m². A rectangle 4 cm by 3 cm has an area of 4 × 3 = 12 cm².
Types of triangles
Triangles are named by their sides (equilateral, isosceles, scalene) or by their angles (acute, right-angled, obtuse). The three angles of any triangle add up to 180 degrees.
Perimeter
Perimeter is the total distance all the way around the outside of a flat shape. For a rectangle 5 cm long and 3 cm wide, it is 5 + 3 + 5 + 3 = 16 cm.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Pi (π)
Pi (π) is the number you get when you divide any circle's circumference by its diameter. It is about 3.14, and its digits go on forever without repeating.
Pythagoras' theorem
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c², where c is the longest side (the hypotenuse). For sides 3 and 4, the hypotenuse is 5.