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Area

Author: Sophia

what's covered
In this lesson, you will learn how to calculate the area of a rectangle, square, and circle using formulas for area. Specifically, this lesson will cover:

Table of Contents

1. Area Formulas

When we try to determine how much space is covered in a two-dimensional space, we need to use the concept of area. For example, when buying a house we oftentimes will consider the total living space in a house, which is considering how much floor space the house has.

Because area is represented in two dimensions, we represent quantities of area as squared units. For example, we might say that a square living room has a total area of 625 square feet, which might mean that the length of the room is 25 feet and the width is 25 feet.

brainstorm
Can you think of another example where we might need to use area?

Suppose we wanted to paint the outer walls of a house, we may want to estimate the total area of the house’s exterior to get an estimate of the total amount of paint that we would need.

big idea
You will notice that when calculating area you will always be multiplying two distances with one another to determine the two-dimensional space covered by an object.

When calculating areas, there are a few common shapes that you will always come across, as we have previously defined. These shapes and their area formulas are listed below.

formula to know
Area of a Rectangle
A equals b h, where A is the area, b is the base, and h is the height.

rectangle with length l and width w

hint
Note that the area of a square is a special case of the area of a rectangle formula where the length and width are the same.

formula to know
Area of a Triangle
A equals 1 half b h, where A is area, b is base, and h is height.

triangle with height h and base b

hint
Note that the height of a triangle is the distance of the line from one vertex (or corner) of the triangle to the opposite base, such that the line is perpendicular to the base.

formula to know
Area of a Circle
A equals pi space r squared

circle with radius r

hint
Note that the radius of a circle is the distance from the center of a circle to the edge of the circle. Pi is a constant irrational number equal to 3.14159265….

term to know
Pi (π)
The ratio of a circle's circumference to its diameter; approximately equal to 3.14.

Let’s look at how we can use these different area formulas to find some unknown quantity.


2. Calculate the Area Given Side Lengths or Radii

EXAMPLE

Suppose we are told that a rectangle has a length of 10 feet and a width of 12 feet. How would we find the rectangle’s area?

Area equals l times w l equals space 10 space ft comma space w equals 12 space ft
Area equals open parentheses 10 space ft close parentheses open parentheses 12 space ft close parentheses Multiply length by width
Area equals 120 space ft times ft Units of feet are also multiplied
Area equals 120 space ft squared Our Solution

Notice that when calculating area, we square the units of distance.

EXAMPLE

Suppose we are asked to find the area of a circle with a diameter of 9 inches. How would we make this calculation? As with the previous example, we start by writing down the appropriate area formula, and substitute in the quantities we know.

Here, we are given the diameter of the circle, but we need to know the radius to use the formula. The diameter of a circle is simply the distance of the line passing through the center of a circle and touching the circle’s edge. In other words, the diameter, d, is twice as long as the radius, r, open parentheses d equals 2 r close parentheses.

Since the diameter for this circle was 9 inches, the radius must be 4.5 inches.

Area equals pi r squared pi almost equal to 3.14 comma space r equals 4.5 space in
Area equals straight pi open parentheses 4.5 space in close parentheses squared Substitute 4.5 in for r
Area equals 20.25 straight pi space in squared Square the radius
Area equals 63.6 space in squared Our Solution, rounded to the tenths place


3. Calculate Side Lengths or Radii Given Area

Sometimes, we may be given the area of an object and need to back-solve to find the measurements of a given part of an object. Let’s look at some examples.

EXAMPLE

If the area of a square is 400 square feet what is the length of the square?

When solving these types of problems, we follow the same process as we did before but then do some algebraic manipulation to solve the problem. Note that the length and width of a square is the same so we can refer to side length using a single variable, s.

400 space sq space ft equals s squared A r e a equals 400 space sq space ft
square root of 400 space sq space ft end root equals square root of s squared end root Take the square root of both sides
20 space ft equals s Our Solution

EXAMPLE

If the area of a circle is 100 straight pi space ft squared comma what is the radius of the circle?

Like in the previous example, we begin by substituting what we know into the appropriate formula. Notice that the area contains pi in it. This is often times the same when we represent area exactly; we leave the pi multiplied to the number.

100 straight pi space sq space ft equals πr squared Area equals 100 straight pi space sq space ft
100 space sq space ft equals r squared Divide both sides by straight pi
square root of 100 space sq space ft end root equals square root of r squared end root Take the square root of both sides
10 space ft equals r Our Solution

summary
As an introduction to area, we learned that area of an object is the amount of space enclosed in a two dimensional shape. Area is measured in square units, such as centimeters squared or inches squared. Common area formulas include areas of rectangles, triangles, and circles. We can also use formulas to calculate the area when given side lengths or radii, as well as to calculate the side lengths or radii when given an area.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
Pi (π)

The ratio of a circle's circumference to its diameter; approximately equal to 3.14.

Formulas to Know
Area of Circle

A subscript c i r c l e end subscript equals pi r squared

Area of Rectangle

A subscript r e c tan g l e end subscript equals b h

Area of Triangle

A subscript t r i a n g l e end subscript equals 1 half b h