Area Worksheet

Because we need to be able to compare areas regardless of what shape or boundary we want to find the area for, we use the same basic unit for all areas.

	Definition	Basic Unit	Notation
Area	Size of the space inside a shape or boundary	Unit square

So essentially, when we find the area of a shape or boundary, we’re finding out how many unit squares fit inside of that shape or boundary. Take a look at the examples below.

In order to understand some of the terms used when describing shapes, you should understand the definitions of some basic concepts.

Area of Rectangle

In order to find the area of a rectangle, let’s first define what a rectangle is. Take a look at the properties of a rectangle below.

All of the properties above have special notations. When you have any right angle, you can mark it as below.

When we talk about equal sides, or congruent sides, we can also mark them with specific notation. Take a look below.

As you can see, the opposite sides are marked with matching dashes, meaning they are congruent. Now let’s take a look at the area of a rectangle. In general, shapes have a length and a width. Take a look at the notation below.

	Notation
Length
Width

The area formula for a rectangle is written below.

	Area Formula
Rectangle

Area of Square

To understand the area of a square, let’s first take a look at the properties of a square.

Now, let’s take a look at the notation of a square.

Four right angles
Four congruent sides

Because all the sides of a square are of equal length, this means that the length and width of a square are equal to each other. Take a look at the area formula of a square below.

	Sides	Area Formula
Square

Area of Circle

Circles are very special shapes in maths. Let’s take a look at the properties of circles in order to better understand how to find the area of a circle.

Property 1	A circle’s length is called a diameter
Property 2	The length from the centre of the circle to the outside is called the radius
Property 3	The length around the circle, or the perimeter, is called the circumference

There are three ways you can find the area of a circle using each of the properties above. Take a look at the formulas below.

Using the diameter
Using the radius
Using the circumference

Area of Equilateral Triangle

While we usually only think of triangles as being the same, there are actually three different types of triangles. These triangles are defined by their properties. Let’s start by looking at an equilateral triangle definition.

Here, we have a new type of notation we haven’t seen before. This notation is used when we have angles that are the same size. Take a look below.

The area of an equilateral triangle can be found in the table below.

	Area
Equilateral Triangle

Area of Any Triangle

While you can find the area of an equilateral triangle with the special formula above, the area of triangles can also be found with a general formula. This formula is written below.

	Area Formula
Any triangle

As you can see, this formula uses several variables we haven’t seen before. Take a look at the table below for a description of these variables.

	Base	The bottom side, or base, of any triangle
	Height	The size of the line that goes from the base to the furthest point away from the base.

Problem 1

Find the area of the circle with the following parameters.

Problem 2

Find the area of the shaded region below.

Problem 3

Find the area of the triangle below.

Solution Problem 1

In this problem, you can find the area of the circle three different ways given the parameters.

Using the diameter
Using the radius
Using the circumference

The area is equal to 28.27.

Solution Problem 2

Here, we have a rectangle that contains a square. In order to find the area of the shaded region, we simply do the following.

Solution Problem 3

Here, we know because of the notation that the sides marked with a dash are equal to each other. To find the shaded region, we simply need to find the area of the following triangle.

Step 1	Use Pythagorean theorem to find the height of the triangle:
Step 2
Step 3
Step 4
Step 5