How to find the area of a triangle

Which method you use depends on what you know about the triangle. This page goes through all of them, from the GCSE basics to the formulas for when you have no height at all.

Why the area is half base times height

Draw a rectangle around a triangle, with the same base and height. The height line splits the triangle into two right-angled triangles, and each one is exactly half of its part of the rectangle. So the triangle is half the rectangle, and a rectangle's area is base × height. That is where the ½ comes from, and it works for any triangle.

How to find the area of a triangle: step by step

1. Pick a base. Any side can be the base, and it is usually the side at the bottom.
2. Find the perpendicular height: the distance from the base to the opposite corner, measured at right angles to the base. A small square on a diagram marks the right angle.
3. Check the units match, for example both in centimetres.
4. Multiply and halve: ½ × base × height.
5. Give square units: cm², m² or mm².

Area of a right-angled triangle

In a right-angled triangle, the two sides that meet at the right angle are already a base and a height. A right-angled triangle with shorter sides of 6 cm and 9 cm has area ½ × 6 × 9 = 27 cm². Don't use the longest side, the hypotenuse, unless you know the height to it.

Area of an obtuse triangle

If the triangle has an angle over 90°, the height can fall outside it. Extend the base with a dashed line and measure the height to that line. The formula doesn't change.

Here the area is ½ × 5 × 4 = 10 cm². Use the base 5 cm, not the base plus the dashed extension.

Worked examples

Example 1: base and height

A triangle has a base of 10 cm and a perpendicular height of 7 cm. Area = ½ × 10 × 7 = 35 cm².

Example 2: finding the height from the area

A triangle has an area of 48 cm² and a base of 12 cm. What is its height? Substitute into the formula: 48 = ½ × 12 × h, so 48 = 6h and h = 8 cm.

Example 3: mixed units

A triangular sail has a base of 2.4 m and a height of 150 cm. Convert first: 150 cm = 1.5 m. Area = ½ × 2.4 × 1.5 = 1.8 m².

Area of a triangle using sine: ½ab sin C

If you know two sides and the angle between them, the included angle, you don't need the height:

Area = ½ × 9 × 7 × sin 40° = 20.25 cm² to 2 decimal places. Make sure your calculator is in degrees.

Why it works: the height from C to the base is 7 × sin 40°, from right-angled trigonometry. Put that into ½ × base × height and you get ½ × 9 × 7 × sin 40°. In GCSE Maths, ½ab sin C is Higher tier, and it is printed on the 2027 Higher formula sheet. ½ × base × height isn't on any sheet, so learn it. You can check answers like this in the Trigonometry Calculator.

Try one yourself, then press Mark to check it in our Trigonometry course.

Area of an equilateral triangle

All three sides of an equilateral triangle are equal, so its area depends only on the side length a:

This comes from ½ab sin C with both sides a and the angle 60°, since sin 60° = √3 ÷ 2. For a side of 6 cm, area = √3 ÷ 4 × 36 = 15.59 cm².

Heron's formula: area from three sides

If you know all three sides but no angles or height, use Heron's formula, named after Heron of Alexandria, who gave a proof of it in the first century AD. First find the semi-perimeter s, half the perimeter:

For sides of 5 cm, 6 cm and 7 cm: s = (5 + 6 + 7) ÷ 2 = 9. Area = √(9 × 4 × 3 × 2) = √216 = 14.70 cm². Heron's formula isn't on the GCSE syllabus, but it is a handy check, and the cosine rule followed by ½ab sin C gives the same answer.

Area of a triangle from coordinates

If you know the coordinates of the three corners, (x₁, y₁), (x₂, y₂) and (x₃, y₃), use:

For P(1, 2), Q(5, 3) and R(2, 7): ½ × |1(3 − 7) + 5(7 − 2) + 2(2 − 3)| = ½ × |−4 + 25 − 2| = ½ × 19 = 9.5 square units. The bars mean take the positive value, since listing the corners the other way round gives a negative. Without the formula, you can draw a rectangle around the triangle on the grid and subtract the three right-angled triangles outside it.

Which method should you use?

Common mistakes

• Forgetting the ½. Base × height on its own is the area of the rectangle, twice the triangle.
• Using a slanted side as the height. The height must meet the base at a right angle.
• Using an angle that isn't between the two sides in ½ab sin C. The angle must be the one where the two known sides meet.
• Calculator in radians. Check it shows D or DEG before using sin.
• Leaving out square units. Area is in cm², m² and so on, not cm.
• Mixing units. Convert so both lengths use the same unit before multiplying.

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