Triangle Area Calculator

How it works

The area of any triangle is half its base times its height, where the height is measured perpendicular to the base. Every other formula here is a way of finding that same area when the height is not something you can measure directly.

With three sides, Heron's formula does it: take half the perimeter s, then the area is the square root of s(s − a)(s − b)(s − c). The three sides must satisfy the triangle inequality, each shorter than the other two together, or no triangle exists. With two sides and the angle between them, the area is half the product of the sides times the sine of the angle, and the law of cosines gives the third side for the perimeter.

An equilateral triangle has height (√3 ÷ 2) × side, so its area is (√3 ÷ 4) × side². An isosceles triangle with two equal legs splits into two right triangles, so its height is √(leg² − (base ÷ 2)²). The height shown is always measured to side a in the three-side and angle modes, to the base in the other modes.

Formula

base and height:  area = ½ × base × height
three sides:      s = (a + b + c) ÷ 2;  area = √(s(s − a)(s − b)(s − c))
two sides + angle: area = ½ × a × b × sin C;  c = √(a² + b² − 2ab cos C)
equilateral:      area = (√3 ÷ 4) × a²;  height = (√3 ÷ 2) × a
isosceles:        height = √(a² − base² ÷ 4);  area = ½ × base × height
perimeter = a + b + c;  height to the base = 2 × area ÷ base

Example

A triangle with sides 13, 14 and 15 ft has s = 21, so the area is √(21 × 8 × 7 × 6) = √7056 = 84 ft². The perimeter is 42 ft and the height measured to the 13 ft side is 2 × 84 ÷ 13 ≈ 12.92 ft.

A triangle with a 10 ft base and a 6 ft height has an area of ½ × 10 × 6 = 30 ft². The base and height alone do not fix the other two sides, so no perimeter is shown.

Assumptions and limitations

• All lengths are in the unit you choose; the area is in that unit squared. The unit labels the result and does not convert anything.
• The included angle is entered in degrees and must lie strictly between 0° and 180°.
• Base and height alone do not determine the perimeter, so that mode leaves it blank.
• Results are rounded for display only.

Frequently asked questions