Solve problems involving a variety of different types of triangles

Triangles are polygons with three sides. The angles between the sides are the interior angles. The total sum of interior angles of any triangle equals 180°. Let’s take a look at some common triangles and their characteristics.

  • A scalene triangle is one in which all three angles are different, therefore all its sides have different lengths.
  • An acute triangle contains three acute angles. In other words, each angle of an acute triangle is less than 90°.
  • An isosceles triangle is one with two equal sides, and the angles making up those two sides are also equal.
  • An equilateral triangle is one with three equal sides and angles.
  • right triangle is one that contains a 90°.

Furthermore, an exterior angle of a triangle is the angle between the side of a triangle and an extension of the adjacent side, such as angle θ. An external angle equals the sum of the two opposite interior angles.

Area of a Triangle

There are several different formulas you can use to find the area of a triangle, but depending on what information about the triangle that’s given, you have to decide which one is best to use.

The first formula shown is used when you know the base and height of the triangle. The height is defined as the distance from any vertex directly to the the opposite side (shown below ↓). The key to using this formula, of course, is to know the altitude (height) of a triangle.

Area of a triangle


Where A is the area, b is the base, and h is the height.

This next formula is used when we know the three sides, but not the altitude.

Hero’s formula (Heron’s formula)


Where A represents the area, a↔c presents each side, and s represents half the perimeter. An example where Heron’s formula is used is shown below:

Similar Triangles

Similar triangles are formed when you have two different triangles both sharing the same three angles. This makes their lengths, while different, proportional to one another. This suggests that similar triangles are not congruent figures, meaning that they’re not identical in both length and angles, as shown below.The first video below will show you how to mathematically identify a pair of similar triangles and explain why they are similar using symbols and letters. You’ll discover that to refer to any angle in a triangle (↓), let’s say angle A (∠A) in the one below – relative to B and C – we write ∠BAC or ∠CAB.

Notice that the angle in reference is always in the middle of the statement.

Pythagorean Theorem

Learning about triangles isn’t complete without knowing the Pythagorean theorem. In a right triangle, the side opposite the right angle is called the hypotenuse, and  the other two sides are called legs. The legs and the hypotenuse are related by the Pythagorean theorem.


A detailed lesson on this matter is shown below: