How to Calculate the Area of a Triangle

By | October 26, 2014

We have learned about the areas of squares, rectangles, circle, parallelogram, and trapezoid. There is one important shape we haven’t discuss: the area of a triangle.

The area of a triangle is  half the product of its base and height. But did you know where did the formula come from? Let us discuss it in this post.

The area of a triangle is related to the area of other shapes, but we are going to relate it to the area of a parallelogram.

Consider the triangle above with base $b$ and height $h$. If we are going to create another triangle congruent to it (congruent means the same size and shape), then we can form a quadrilateral by coinciding their  two corresponding sides. What is interesting is that every time we do this, we create a parallelogram.

Now, we have already learned that getting the area of a parallelogram $A_p$ is multiplying the base $b$ and the height $h$. So,

$A_p = bh$.

However, in calculating the area of the parallelogram above, we actually calculated for the areas of two triangles. This means that  to get the area of the original triangle, we have to divide by 2.

Therefore, the area of a triangle is

$A = \displaystyle \frac{bh}{2}$.

That is the reason behind the formula.  Now let’s have a worked example.

Example 1

Find the area of a triangle with base 12 units and height 8 units.

Solution

From the derivation above,

$A = \displaystyle \frac{bh}{2}$

$A = \displaystyle \frac{(12)(8)}{2}$

$A = 48$ square units.

Example 2

What is the height of a triangle with area 14 square units and base 4 units.

Solution

Again, we use the original formula above and substitute the values?

$A = \frac{bh}{2}$

$14 = \frac{4h}{2}$

$14 = 2h$

Dividing both sides by 2 gives us

$h = 7$

In the next post, we are going to discuss more examples on problems involving area of triangles.