How triangle area is worked out
Every method finds the same thing: half of the parallelogram the triangle makes with a copy of itself.
A = ½bh · A = √(s(s − a)(s − b)(s − c)) · A = ½ab sin C
- Base and height: A = ½ × base × height, with the height at right angles to the base.
- Three sides: Heron's formula, s = (a + b + c) ÷ 2 and A = √(s(s − a)(s − b)(s − c)). It is computed in a stable form so thin triangles stay accurate.
- Two sides and the angle between: A = ½ × a × b × sin C.
- Three points: the shoelace formula, half the absolute value of x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂).
Worked example — Sides 5, 6 and 7: s = 9, so A = √(9 × 4 × 3 × 2) = √216 = 14.6969.
Common triangles
| Base | Height | Area |
|---|---|---|
| 10 cm | 5 cm | 25 cm² |
| 6 cm | 4 cm | 12 cm² |
| 12 cm | 9 cm | 54 cm² |
| 3 cm | 4 cm | 6 cm² |
| 8 cm | 8 cm | 32 cm² |
| 15 cm | 7 cm | 52.5 cm² |
Swipe the table sideways to see the rest.
From base and height, in centimetres. Each row is worked out by this calculator.
Things to consider
- The height must meet the base at a right angle. A sloping side is not the height.
- With three sides, the longest must be shorter than the other two added together.
- Use the angle between the two sides you know, not one of the others.
- For a right triangle, the two legs are the base and height. The Pythagorean Theorem Calculator: Find the Hypotenuse finds a missing side.
Frequently asked questions
What is the formula for the area of a triangle?
Area = ½ × base × height. A base of 10 and height of 5 gives 25.
What is Heron's formula?
With three sides, take s = (a + b + c) ÷ 2. The area is √(s(s − a)(s − b)(s − c)). Sides 3, 4, 5 give 6.
How do I find the area with two sides and an angle?
Area = ½ × a × b × sin C, where C is the angle between a and b. Sides 5 and 7 with 30° give 8.75.
How do I find the area from coordinates?
Use the shoelace formula: half of |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If it is 0, the points are in a straight line.
What is the height of a triangle?
The perpendicular distance from a corner to the opposite side. In an obtuse triangle it can fall outside, on the extended base.
What is the area of an equilateral triangle?
√3 ÷ 4 × a². A side of 2 gives 1.7321. For right isosceles triangles, see the 45-45-90 Triangle Calculator.
Results are estimates for general information only, not financial, medical or professional advice.