Triangle basics | Geometry | Quantitative reasoning

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

General triangle equations

Let’s start with some 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:

Right triangle with sides of lengths 6 and 8

What is the area of the above triangle?

(spoiler)

Answer: $24$

$A A A A = bh /2 = 8 \times 6/2 = 48/2 = 24$

Keep in mind that the height of the triangle does not necessarily have to be one of the sides - it is simply the height from top to bottom, perpendicular to the base. To illustrate, the height of the triangle drawn below is $3$ , even though you can see that there are no sides with a length of $3$ .

Example triangle area calculation with height and base

Can you calculate the area of this triangle above?

(spoiler)

Answer: $15$

$A A A A = bh /2 = 10 \times 3/2 = 30/2 = 15$

Equation for the perimeter of a triangle

The second most important aspect of a triangle is the perimeter. The perimeter is the sum of all sides, as if you were walking around the outside edge all the way around and back to your starting point.

$P = a + b + c$

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

Right triangle with sides of lengths 6, 8, and 10

We’ve added in the hypotenuse of this triangle from before to make it a little simpler. What is the perimeter of this triangle?

(spoiler)

Answer: $24$

$P A A = a + b + c = 6 + 8 + 10 = 24$

Coincidentally, the area and perimeter for this triangle are both $24$ .

Types of triangles

Although all triangles have three sides, there are a few different types of fundamental triangle shapes.

Equilateral triangles

Equilateral triangles are often the easiest to work with. In an equilateral triangle, the side lengths are all the same, and the inner angles are also all the same. Since the sum of the interior angles of a triangle must be $180$ , each of the three angles is $180/3 = 60$ degrees. And since the sides and angles are all the same, if you know the length of one side of an equilateral triangle, then you know the length of the others too.

Example of an equilateral triangle

Sidenote

Congruence marks

Did you notice those short lines that intersect the sides of the triangle? Those are called congruence marks (a.k.a. hatch marks or hash marks), and they mean that the sides are the same length.

You might see these marks on sides or angles, and there might be several with a different number of lines. You just match them up! The sides/angles with one mark will be identical to the others with one mark, the sides/angles with two marks will be identical to the others with two marks, and so on.

Right triangles

Right triangles will always have a 90-degree angle. The hypotenuse (i.e. the longest side) of a right triangle will always be the side opposite to this right angle.

Example of a right triangle

The area of a right triangle is easy to calculate because the two non-hypotenuse legs are always the height and base. Furthermore, if you know any two sides of a right triangle, you can solve for the missing side using the Pythagorean Theorem, which we’ll cover later.

Isosceles triangles

Isosceles triangles have one side and one angle “isolated” from the others. In other words, two angles and two sides are identical, while the third side and angle are different. If you know you’re working with an isosceles triangle and are given two of the angles, you can calculate the missing one: $∠ C = 180 - ∠ A - ∠ B$ .

Example of an isosceles triangle

Scalene triangles

Scalene triangles have no identical side lengths or angles, and are the most irregular triangles. Basically, if a triangle isn’t one of the types above, then it’s a scalene triangle.

For these, you’ll need to take all the information you can from the problem in order to solve for any specific measurements. The most common question about scalene triangles involves the Triangle inequality theorem, a.k.a. the Third side rule, which we will also discuss in a later chapter.

Example of a scalene triangle