To find the area of any triangle, multiply the base by the perpendicular height and halve the result: A = ½ × b × h. A triangle with a base of 10 cm and a height of 6 cm has an area of 30 cm².
The formula is short, and most mistakes come from the height, not from the multiplication. This guide shows why the formula works, how to choose the base and the height, what changes for right, isosceles, obtuse and equilateral triangles, the ½ab sin C formula used at GCSE Higher, and the slips that cost marks.
The triangle area formula, and why it works
Draw a rectangle around a triangle, with the same base and the same height. The triangle covers exactly half of it, which is where the half comes from: the rectangle has an area of base × height, so the triangle has an area of ½ × base × height.
The same idea works for any triangle. Two copies of the same triangle, one turned round, fit together into a parallelogram with the same base and height, and a parallelogram has the same area as the rectangle you get by cutting off one end and moving it to the other. In England, the national curriculum for mathematics asks pupils in Year 6 to calculate the area of parallelograms and triangles, and its guidance suggests exactly this route: relating the area of rectangles to parallelograms and triangles by dissection, then using the formula in words or symbols. At Key Stage 3, pupils derive and apply the formula to solve problems.
In the United States, the Common Core standards place the same skill in grade 6: finding the area of right triangles and other triangles by composing them into rectangles or decomposing them into triangles.
Choosing the base and the perpendicular height
Any side of a triangle can be the base. The height that goes with it is the perpendicular distance from that base to the opposite corner, measured at a right angle to the base, never along a sloping side. Each triangle has three possible bases and three matching heights, and all three pairs give the same area.
| Triangle | Base | Perpendicular height | Area |
|---|---|---|---|
| Right triangle, legs of 6 cm and 8 cm | 6 cm | 8 cm | 24 cm² |
| Isosceles, base 10 cm, equal sides 13 cm | 10 cm | 12 cm | 60 cm² |
| Obtuse, height outside the triangle | 6 cm | 4 cm | 12 cm² |
| Equilateral, sides of 10 cm | 10 cm | about 8.66 cm | about 43.3 cm² |
| Garden bed | 3.5 m | 2 m | 3.5 m² |
The answer is always in square units: cm², m² or square feet. If the base and the height are in different units, convert one of them first. A base of 2 m and a height of 50 cm is 2 m and 0.5 m, so the area is ½ × 2 × 0.5 = 0.5 m², and certainly not 50. Our guide to converting units covers the conversions between metric and imperial measures.
The area of a right triangle
A right triangle is the easiest case, because the two sides that meet at the right angle are already a base and a perpendicular height. Multiply them and halve: legs of 6 cm and 8 cm give 6 × 8 = 48, and half of 48 is 24 cm².
Ignore the longest side, the hypotenuse, which is 10 cm in this example. It is never the height of a right triangle, since it is not perpendicular to either of the other sides. If one leg is missing, Pythagoras' theorem finds it from the other two sides: with a hypotenuse of 13 cm and a leg of 5 cm, the other leg is the square root of 169 − 25 = 144, which is 12 cm, and the area is ½ × 5 × 12 = 30 cm².
The same trick gives the height of an isosceles triangle. The height from the top corner cuts the base in half and makes two right triangles. With a base of 10 cm and equal sides of 13 cm, each half has a base of 5 cm and a hypotenuse of 13 cm, so the height is 12 cm and the area is ½ × 10 × 12 = 60 cm².
How to find the area of an obtuse or an equilateral triangle
In an obtuse triangle, one angle is wider than 90°, and the height from one of the corners falls outside the triangle. Extend the base with a straight line, and measure the height from the top corner down to that extended line, at a right angle. The base itself does not change: if it is 6 cm and the height to the extended line is 4 cm, the area is ½ × 6 × 4 = 12 cm². Only the part of the base that belongs to the triangle counts, never the extension.
For an equilateral triangle, every side is the same, and the height cuts it into two right triangles with a base of half a side. With sides of 10 cm, the height is the square root of 100 − 25 = 75, about 8.66 cm, and the area is about ½ × 10 × 8.66 = 43.3 cm². For any equilateral triangle with side a, this simplifies to a² × √3 ÷ 4, or about 0.433 × a².
When the drawing gives no height at all, look for a right angle somewhere in the figure, or split the shape into right triangles you can solve one at a time.
With two sides and the angle between them
Sometimes a problem gives two sides and the angle between them, without the height. Then the area is ½ × a × b × sin C, where a and b are the two sides and C is the angle they make. In the AQA GCSE Mathematics specification, this formula is Higher tier content, and in a normal year it is one of the formulae students learn by heart. For now, though, Ofqual requires exam boards to give students a formulae sheet with the formulae they would normally memorise, ½ab sin C among them, and it has decided to keep the sheet for the lifetime of the current GCSE specifications. Knowing it without looking still saves time.
A worked example: sides of 8 cm and 5 cm with an angle of 30° between them. Since sin 30° = 0.5, the area is ½ × 8 × 5 × 0.5 = 10 cm². With an angle of 90°, sin 90° = 1 and the formula becomes ½ × a × b, which is the right triangle rule again: the two formulas agree.
At Foundation tier, the same specification asks students to know and apply the formula for the area of triangles, parallelograms and trapezia. A rectangle is base × height, a parallelogram is base × perpendicular height, a trapezium is half the sum of the parallel sides times the height, and a triangle is half of base × height. The family resemblance makes them easier to remember together, and the method for the curved shapes is in our guide to the area of a circle.
Common slips, and quick checks in your head
Most lost marks come from a handful of mistakes, always the same ones.
| Slip | What it looks like | The fix |
|---|---|---|
| Forgetting the half | 14 × 9 = 126 cm² | ½ × 14 × 9 = 63 cm² |
| Using a sloping side as the height | ½ × 10 × 13 = 65 cm² | Use the perpendicular height: ½ × 10 × 12 = 60 cm² |
| Mixing units | ½ × 2 × 50 = 50 | Convert first: ½ × 2 × 0.5 = 0.5 m² |
| Wrong unit in the answer | 63 cm | An area is in square units: 63 cm² |
| Measuring the extended base | Adding the extension to the base | Keep the triangle's own base |
In your head, halve the even number first. ½ × 14 × 9 is 7 × 9 = 63, and ½ × 15 × 8 is 15 × 4 = 60. If both numbers are odd, multiply and then halve: ½ × 7 × 9 is 63 ÷ 2 = 31.5. A quick estimate also catches the worst slips: the triangle must be smaller than the rectangle around it, so its area is always less than base × height. More shortcuts of this kind are in our mental maths tricks.
Frequently asked questions
What is the formula for the area of a triangle?
Half of the base times the perpendicular height: A = ½ × b × h. With a base of 10 cm and a height of 6 cm, the area is 30 cm².
How do you find the height of a triangle?
It is the perpendicular distance from the base to the opposite corner. In a right or isosceles triangle, Pythagoras' theorem gives it from the sides.
Can any side be the base of a triangle?
Yes. Each side can be the base, as long as you use the height that is perpendicular to that side. All three pairs give the same area.
Which units do you use for the area of a shape?
Square units: cm², m², square inches or square feet. Convert the base and the height to the same unit before multiplying.
When do you use ½ab sin C?
When you know two sides and the angle between them, without the height. At GCSE it is Higher tier content, and it appears on the formulae sheet that exam boards must give students.
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