How to Find the Area of a Triangle

Although all triangles have three sides, not all triangles are exactly the same. Here are four different types and additional ways to find the area for each one.

1. Right Triangle

A right triangle has one angle equal to 90 degrees. The two sides that form this right angle are the base b and the height h. To find the area, you can use:

In a right-angled triangle, the relationship between the sides and the angles is fundamental. The side opposite the right angle is called the hypotenuse, and it is the longest side of the right-angled triangle. The other two sides are known as the legs.

For example, in a right-angled triangle with legs measuring 3 units and 4 units, the hypotenuse can be found using the Pythagorean theorem.

2. Isosceles Triangle

An isosceles triangle has two sides of the same length and two angles of the same measure. If the equal sides have length a, and the base has length b, you can apply the same triangle area formula you might use for a right-angled triangle.

First, you determine the height h, which you can calculate using the Pythagorean theorem, splitting the base into two equal parts:

Now use the triangle area formula to find A:

By plugging the known dimensions into this triangle area formula, you can calculate the area of the isosceles triangle accurately.

3. Equilateral Triangle

An equilateral triangle has all sides equal and all angles equal to 60 degrees. This symmetry means you have to use a special formula to calculate the area. For an equilateral triangle with side length a, you can calculate the area using:

Being perfectly symmetrical, an equilateral triangle is a fundamental shape in both geometry and the real world. Whether you’re looking to create a perfect aesthetic, such as in designing a triangular garden plot, or attempting to distribute weight evenly, such as in the construction of a truss for a bridge, the principles governing the area of an equilateral triangle can be valuable.

4. Scalene Triangle

To find the area of a scalene triangle, where all the sides have different lengths, you could use a tool called Heron’s formula:

First, you calculate the semi-perimeter, or half the sum of all sides. Mathematicians refer to this figure as s. To find the semi-perimeter, you use:

Now continue with Heron’s formula to find the area A.