How to Find the Area of a Triangle

Child wearing pink headphones, holding a stylus and using a tablet at a home desk — You'll use the same formula to find the area of a right triangle, acute triangle, scalene triangle, equilateral triangle and any other three-sided shape you can come up with. MTStock Studio / Getty Images

With its three sides and three angles, the triangle is one of the most basic shapes in geometry. This means calculating the area of a triangle is a fundamental skill in geometry, with multiple formulas available depending on the type of triangle and the given data.

But knowing how to find the area of a triangle has plenty of applications beyond mathematics. For example, understanding right-angled triangles is essential for finding accurate measurements in construction and navigation. Isosceles triangles are crucial in structural engineering, aerospace design, and optics, where precision is absolutely a requirement. And understanding the area of an equilateral triangle is essential in architecture and even art.

Let's review the basic formula for finding the area of a triangle, plus formulas for other scenarios.

Basic Formula for Area

The most basic triangle formula for finding area involves the shape's base and height. For any triangle with a base b and a height h, you can find the area A by using:

A = 1/2 × b × h

This formula represents half the product of the base and the height, which is the perpendicular drawn from the base to the opposite vertex.

4 Types of Triangles and Their Formulas

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:

A = (1/2) x b x h

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.

c = √(a² + b²)

c = √(3² + 4²)

c = √(9 + 16)

c = √25

c = 5 units

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:

h = √[a² – (b/2)²]

Now use the triangle area formula to find A:

A = (1/2) x b x h

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:

A = [(√3)/4] x a²

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:

A = √[s x (s – a) x (s – b) x (s – c)]

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:

s = [(a + b+ c)/2]

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

Using Heron's Formula

Heron's formula is a powerful tool in geometry that allows you to calculate the area of a triangle when you know the lengths of all three sides:

A = √[s x (s – a) x (s – b) x (s – c)]

Named after the ancient Greek mathematician Hero of Alexandria, this formula is particularly useful because it does not require you to know the height of the triangle or any of its angles opposite the given sides.

Suppose you have a scalene triangle with the following side lengths:

a = 5
b = 6
c = 7

First, you determine the semi-perimeter, or s. Again, the formula for semi-perimeter is:

s = [(a + b+ c)/2]

Calculating the semi-perimeter in this example looks like this:

s = [(5 + 6+ 7)/2] = 9

Now, you find the area A using:

A = √[s x (s – a) x (s – b) x (s – c)]

To apply this formula to the above example, you make the following calculations:

A = √[9 x (9 – 5) x (9 – 6) x (9 – 7)]

A = √(9 x 4 x 3 x 2)

A = √216

A = approximately 14.7 square units

Advantages of Heron's Formula

Mathematicians have used Heron's formula for centuries because it offers certain advantages:

It's versatile. Heron's formula works for all types of triangles, whether they are scalene (no equal sides), isosceles (two equal sides), or equilateral (all sides equal).
You don't need to know the height or angles. Unlike other triangle area formulas that require the height or specific angles, Heron's formula only needs the side lengths.
It results in an exact figure. Heron's formula provides an exact area measurement without the need for additional geometric constructions.

Connection to Other Triangle Formulas

Heron's formula complements other methods of finding a triangle's area. For example, while the basic triangle formula — A = (1/2) x base x height — is straightforward for right triangles or when you know the height, Heron's formula provides a way to calculate the area solely from side lengths. You could use both methods to be confident in your resulting figures.

Other Formulas for Specific Cases

Sometimes a scenario calls for a special formula, such as the following cases.

SAS (Side-Angle-Side)

When you know the two sides and the included angle of a triangle, you can calculate the area using the formula:

A = (1/2) x a x b x sin(C)

where a and b are the sides, and C is the included angle.

ASA (Angle-Side-Angle)

For a given triangle where you know two angles and the included side, you can find the area using:

A = [a² x sin(B) x sin(C)]/[2 x sin(A)]

where a is the known side, and A, B and C are the angles.

We created this article in conjunction with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.