Area of a triangle
The area of a triangle is the region that is bounded by its three sides.
![](/img/a/geometry/triangles/area-of-a-triangle/area-of-a-triangle.png)
The area of a triangle can be found using a number of different methods.
Using base and height
Given base b and height h for a triangle, the area of the triangle is:
The height of a triangle is perpendicular to its base. It can be either inside, outside, or on the side as shown in the figure below.
![](/img/a/geometry/triangles/area-of-a-triangle/base-and-height.png)
The formula above can be proven using the area of a parallelogram. The formula for the area of a parallelogram is base × height, written as Area = b × h. A diagonal of a parallelogram divides it into two congruent triangles as shown in parallelogram ABCD below.
![](/img/a/geometry/triangles/area-of-a-triangle/parallelogram.png)
Since △ABC≅△CBD, the area of △ABC is half the area of the parallelogram. Therefore, the area of △ABC=. Similarly, the area of △CBD=
.
Using its sides
If the three sides of a triangle are given, the area can be found using Heron's formula.
where a, b, and c are the lengths of the three sides of the triangle, the semi-perimeter, s, is .
![](/img/a/geometry/triangles/area-of-a-triangle/herons-formula.png)
One way to prove Heron's formula is by using the Pythagorean Theorem. In the figure above, we have d2 + h2 = c2 and (b - d)2 + h2 = a2. Subtracting these we get 2bd = b2 + c2 - a2. So,
Since h2 = c2 - d2
h2 = | |
= | |
= | |
= | |
= | |
= |
Area = | |
= | |
= |
Using trigonometry
If two sides and an included angle are given for a triangle, we can find its area using the following formulas:
where A, B, and C are the 3 angles of the triangle and a, b, and c are the lengths of the 3 sides opposite the 3 angles.
![](/img/a/geometry/triangles/area-of-a-triangle/area-using-trigonometry.png)
For the triangle shown above with base b, height h, sides a, and angle C given, we can use the sine function to get and find that h = a × sin(C).
Using Area = .
Example:
For △ABC, a = 7, b = 4, and ∠C=40°. Find the area of the triangle.
Since we are given two sides and an included angle we can use the following formula to get,
Area = | |
= | 14 × 0.6428 |
= | 8.9992 |