Area of a Triangle

In the case where you have non-right-angled triangles, different methods are required to figure out the area and angles of the triangle.

Right-angled triangles

area of triangle 1


In the simple case of a right-angled triangle, the area is given by:

area = \frac{1}{2}bh

Using the sin rule

If given the lengths of two sides of a triangle and also the angle joining the two sides the sine rule can be applied to determine the area of a triangle. For a triangle ABC with angles a,b & c shown below:


If lengths and b are known and the angle A is known then the area of the triangle is:

area = \frac{1}{2}bc \sin{A}

Heron’s formula

triangle notation

Given three lengths of a triangle, the area of that triangle can be determined. In Heron’s formula a special variable s is introduced. is the semi-perimeter and is equal to:

s = \dfrac{a + b + c}{2}

The area of the triangle is therefore:

area = \sqrt{s(s - a)(s - b)(s - c)}


Find the area of the following triangle:

herons formulaFirst, find the value of s

s = \dfrac{a + b + c}{2}
s = \dfrac{6+5+8}{2}
s = 9.5

Then, use Heron’s formula
area = \sqrt{s(s - a)(s - b)(s - c)}
area = \sqrt{9.5(9.5-6)(9.5-5)(9.5-8)}
area = 15 units^2

See also: