How do you find the area of a triangle?

The simplest way is half the base times the perpendicular height. If you only know the three side lengths, use Heron's formula instead, which works from the sides alone.

What is Heron's formula?

Heron's formula finds the area from the three sides. Work out the semi-perimeter s, half the sum of the sides, then take the square root of s times s minus a, times s minus b, times s minus c.

Do the three sides always form a triangle?

Only if each side is shorter than the sum of the other two, which is the triangle inequality. If not, no triangle exists and the calculator will say so.

Area of a Triangle Calculator

This calculator finds the area of a triangle in the two ways the information usually arrives. The first and simplest is from the base and the perpendicular height, where the area is just half the base times the height. The second is Heron's formula, which you reach for when you know the three side lengths but not the height, a common situation in surveying, construction and land measurement where you can measure the edges of a triangular block but not its internal height. To use Heron's formula the calculator first finds the semi-perimeter, which is half the sum of the three sides, then takes the square root of the semi-perimeter multiplied by its difference from each side in turn. Before doing so it checks the triangle inequality, the rule that each side must be shorter than the sum of the other two, because three lengths that break this rule cannot close into a triangle and have no area. Triangular areas come up when working out a garden bed, a gable end, a section of roof, an offcut of material or a piece of land, so being able to switch between the base and height method and the three sides method is genuinely useful. Choose your method and enter the measurements to see the area instantly.

Method

Base

Height

Side c

area (square units)

How it works

With base and height, the area is half the base times the height. With three sides, the calculator uses Heron's formula: find the semi-perimeter s as half the sum of the sides, then the area is the square root of s times s minus a, times s minus b, times s minus c. It first checks the three sides can form a triangle.

Worked example

A triangle with a base of 10 and a height of 6 has an area of half of 10 times 6, which is 30 square units. A triangle with sides 3, 4 and 5 has a semi-perimeter of 6 and an area of the square root of 6 times 3 times 2 times 1, which is 6.

Related calculators

Triangle Calculator: solve sides and angles.
Area Calculator: other shapes.
Hypotenuse Calculator: right triangles.
Pythagorean Theorem Calculator: a squared plus b squared.

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