Triangle basics

In this chapter, you’ll learn the basic types of triangles and the equations for area and perimeter.

General triangle equations

Let’s start with equations that work for any type of triangle.

Equation for the area of a triangle

The area of a triangle is half the base times the height.

$$A=bh/2$$

Here’s an example:

What is the area of the triangle above?

Keep in mind that the height of a triangle doesn’t have to be one of its sides. The height is the perpendicular distance from the base to the opposite vertex.

To illustrate, the height of the triangle drawn below is $3$, even though none of the sides has length $3$.

Can you calculate the area of the triangle above?

Equation for the perimeter of a triangle

Another key measurement is the perimeter. The perimeter is the total distance around the triangle, found by adding the lengths of all three sides.

$$P=a+b+c$$

Let’s use the same triangle from earlier as an example.

We’ve added the hypotenuse to complete the triangle. What is the perimeter of this triangle?

Types of triangles

All triangles have three sides, but they can be classified into a few fundamental types.

Equilateral triangles

Equilateral triangles are often the simplest to work with. In an equilateral triangle:

• All three side lengths are equal.
• All three interior angles are equal.

Since the interior angles of any triangle add up to $180^{\circ }$, each angle in an equilateral triangle is $180/3=60^{\circ }$. Because all sides are equal, if you know the length of one side, you know the length of the other two as well.

Right triangles

A right triangle has one $90^{\circ }$ angle. The hypotenuse (the longest side) is always the side opposite the right angle.

The area of a right triangle is especially easy to calculate because the two legs (the sides that are not the hypotenuse) can serve as the base and height.

Also, if you know any two sides of a right triangle, you can find the third side using the Pythagorean Theorem, which you’ll cover later.

Isosceles triangles

In an isosceles triangle, two sides are equal, and the angles opposite those sides are equal. That means two angles and two sides are identical, while the third side and third angle are different.

If you’re given two angles in an isosceles triangle, you can find the third using the triangle angle sum:

$\angle C=180-\angle A-\angle B$.

Scalene triangles

A scalene triangle has no equal side lengths and no equal angles. In other words, if a triangle isn’t equilateral, isosceles, or right (as a defining feature), it may be scalene.

For scalene triangles, you typically need to use all the information given in the problem to find specific measurements. A common topic is the Triangle inequality theorem, a.k.a. the Third side rule, which you’ll discuss in a later chapter.