This example derives the area of a triangle formula by examining the special case of a right triangle and then the general case of a triangle.
Table of Contents
A right triangle is a triangle where one of the angles is equal to 90 degrees.
A rectangle can be formed by drawing two congruent right triangles like this.
The area of this rectangle is equal to two times the area of the right-triangle. We also know that the area of the rectangle is equal to the base multiplied by the height[1] so we can set the two equal to each other.
Solve for the area of the right triangle by dividing both sides by .
The area of a right triangle is equal to one-half the base multiplied by the height.
For the general triangle defined by the base and height , there are two cases to consider. For the first case, the vertex of the triangle that is not part of the base is above the base.
For the second case, this vertex extends beyond the base to either side.
When this vertex is above the base, divide the triangle into two right triangles by drawing a line from the vertex to the base so that the triangle is divided into two right triangles.
Represent the area of the triangle as the sum of the areas of the two right triangles.
Factor out from both expressions to get the following expression.
Then, because , substitute into the equation.
The area of a triangle is equal to one-half the base multiplied by the height.
When the vertex extends beyond the base to either side, we can represent the area of the triangle as the area of the right triangle defined by the base and a height of minus the area of the right triangle defined by the base and a height of .
When we factor out from both expressions, we get the following expression.
The terms cancel and we are left with the following expression.
The area of a triangle is equal to one-half the base multiplied by the height.
-
Area of Rectangle