# A Guide to the GRE/Areas of Triangles

## Areas of Triangles[edit]

The area of a triangle equals half of the base multiplied by the height.

In a right triangle, the base and the height will be the two smallest sides.

The area of this triangle equals (6)(8) divided by 2, or 24, because in a right triangle the base and the height are the two smaller sides

Otherwise, however, the height of the triangle will have to be known or deduced.

The height of an isosceles triangle can be deduced using the Pythagorean Theorem.

Take the original triangle. Split the base in half. The height will be perpendicular.

Under the Pythagorean Theorem:

The larger side squared equals the sum of the other sides squared.

Expand the exponents.

Subtract 25 from both sides.

Take the square root of both sides. The triangle has a height of 12 and thus an area of 30.

### Practice[edit]

1.

What is the area of the triangle?

2.

What is the area of this triangle expressed in terms of s and t

3.

What is the area of the triangle?

### Answers to Practice Questions[edit]

1.

The base of an isosceles triangle can be split in half to form right triangles and calculate the height. In this case, the height equals or The area of the triangle is half of the base multiplied by this amount, or

2.

The area of a right triangle equals half of the product of its two smaller sides. The first of these sides equals t; the second must calculated using the Pythagorean Theorem as The area is thus

3.

This can be solved using the same procedure for isosceles triangles used in the first question. However, there is an even simpler way. An equilateral triangle forms 30º-60º-90º triangles when split down the middle. The sides of a 30º-60º-90º have the proportions 1 -- 2. Thus, the height of this triangle equals and its area equals or