Triangle Problems

No matter what type of triangle you have, there are certain properties that all triangles have. Take a look at the most important property below.

The three types of triangles and their properties can be found below.

Perimeter of a Triangle

The perimeter can be found for any shape. The definition of a perimeter and its notation can be found below.

	Definition	Notation
Perimeter	The sum of each side of the shape

The perimeter of a triangle depends on what type of triangle you’re dealing with. Take a look at the formulas below.

	A	B	C
Triangle	Equilateral	Isosceles	Scalene
Perimeter Formula

Shapes have special notation for sides that are of equal length, which is called congruent. Congruent sides are marked by dashes of the same number. Take a look at the example below.

4 = 4	6

Area of a Triangle

The area of any shape can be defined as the number of base units that can be contained inside of that shape’s boundary. Take a look at the example below.

Base Unit	Notation of Area	Definition	Example
1 square		How many unit squares fit inside	4 unit squares

The area of a triangle can be found in a variety of ways. In general, there are two equations you can use to find the area of a triangle, depending on the type of triangle you have.

Type	Area Formula
Any triangle
Equilateral triangle

The notation of the variables in the formulas above can be explained by the following image.

Any side	Base of the triangle	Height of the triangle

Pythagorean Theorem

The Pythagorean theorem is one of the most important theorems of mathematics. The definition of the theorem can be seen in the table below.

	Pythagorean Theorem
When can it be used?	When you have any right triangle
What does it say?	The sum of the square of the lengths of the two smaller sides equal the square of the longer side
What is the formula?
Why is it important?	You can find the length of any side of a right triangle if you have the lengths of the other two sides

Problem 1

Find the perimeter of an equilateral triangle given the following information.

Equilateral Triangle

Problem 2

Find the perimeter of the triangle formed by the diagonal of the following square.

Problem 3

Find the area of the following triangle.

Problem 4

Find the perimeter of the shaded region.

Solution 1

Here, because we have an equilateral triangle, we know that all the sides are equal to each other. Since we have the length of one side, we can easily find the parameter.

Equilateral Triangle

Solution 2

Here, we have to find the perimeter of the triangle formed by the diagonal of the square. Here, we have to use the properties of a square.

Since a square is made of four right angles, the triangle formed by the diagonal is a right angle. Here, we can use the Pythagorean theorem to find the perimeter.

Step 1	Use the formula of the Pythagorean theorem
Step 2	Rearrange to find b
Step 3	Solve for b
Step 4	Plug in the numbers
Step b	Add the sides to get the perimeter

Solution 3

In order to find the area of the equilateral triangle, you should keep the properties of the equilateral triangle in mind.

This translates into the following.

Step 1	Use the formula of the Pythagorean theorem
Step 2	Rearrange to find a
Step 3	Solve for a
Step 4	Plug in the numbers
Step 5	Use the area formula

Solution 4

Here, we have to find the perimeter of the shaded region. Follow the following steps.