Find the Circumference and Area of Circles

The properties of circles have been studied for over [latex]2,000[/latex] years. All circles have exactly the same shape, but their sizes are affected by the length of the radius, a line segment from the center to any point on the circle. A line segment that passes through a circle’s center connecting two points on the circle is called a diameter. The diameter is twice as long as the radius. The distance around a circle is called its circumference.

 

Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by Greek letter [latex]\pi[/latex] (pronounced “pie”). We approximate [latex]\pi[/latex] with [latex]3.14[/latex] or [latex]\Large\frac{22}{7}[/latex] depending on whether the radius of the circle is given as a decimal or a fraction.

If you use the [latex]\pi[/latex] key on your calculator to do the calculations in this section, your answers will be slightly different from the answers shown. That is because the [latex]\pi[/latex] key uses more than two decimal places.

properties of circles

 

• [latex]r[/latex] is the length of the radius
• [latex]d[/latex] is the length of the diameter
• [latex]d=2r[/latex]
• Circumference is the perimeter of a circle. The formula for circumference is [latex]C=2\pi r[/latex]
• The formula for area of a circle is [latex]A=\pi {r}^{2}[/latex]

Since the diameter is twice the radius, another way to find the circumference is to use the formula [latex]C=\pi \mathit{\text{d}}[/latex]. Suppose we want to find the exact area of a circle of radius [latex]10[/latex] inches. To calculate the area, we would evaluate the formula for the area when [latex]r=10[/latex] inches and leave the answer in terms of [latex]\pi[/latex].

[latex]\begin{array}{}\\ A=\pi {\mathit{\text{r}}}^{2}\hfill \\ A=\pi \text{(}{10}^{2}\text{)}\hfill \\ A=\pi \cdot 100\hfill \end{array}[/latex]

We write [latex]\pi[/latex] after the [latex]100[/latex]. So the exact value of the area is [latex]A=100\pi[/latex] square inches. To approximate the area, we would substitute [latex]\pi \approx 3.14[/latex].

[latex]\begin{array}{ccc}A& =& 100\pi \hfill \\ \\ & \approx & 100\cdot 3.14\hfill \\ & \approx & 314\text{ square inches}\hfill \end{array}[/latex]

Remember to use square units, such as square inches, when you calculate the area.

Circumference and Area

Select Video:How to Calculate the Circumference of a CircleCircles – Area, Circumference, Radius & Diameter Explained!How to Calculate Circumference of a Circle (Step by Step) – Circumference Formula

You can view the transcript for “How to Calculate the Circumference of a Circle” here (opens in new window).

You can view the transcript for “Circles – Area, Circumference, Radius & Diameter Explained!” here (opens in new window).

You can view the transcript for “How to Calculate Circumference of a Circle (Step by Step) | Circumference Formula” here (opens in new window).

A circular sandbox has a radius of [latex]2.5[/latex] feet. Find the:

1. Circumference of the sandbox
2. Area of the sandbox

Show Solution

1. Circumference of the sandbox
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the circle
Step 3. Name. Choose a variable to represent it.	Let c = circumference of the circle
Step 4. Translate. Write the appropriate formula Substitute	[latex]C=2\pi r[/latex] [latex]C=2\pi \left(2.5\right)[/latex]
Step 5. Solve the equation.	[latex]C\approx 2\left(3.14\right)\left(2.5\right)[/latex] [latex]C\approx 15\text{ft}[/latex]
Step 6. Check. Does this answer make sense?	Yes. If we draw a square around the circle, its sides would be [latex]5[/latex] ft (twice the radius), so its perimeter would be [latex]20[/latex] ft. This is slightly more than the circle’s circumference, [latex]15.7[/latex] ft.
Step 7. Answer the question.	The circumference of the sandbox is [latex]15.7[/latex] feet.

2. Area of the sandbox
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the circle
Step 3. Name. Choose a variable to represent it.	Let A = the area of the circle
Step 4. Translate. Write the appropriate formula Substitute	[latex]A=\pi {r}^{2}[/latex] [latex]A=\pi{\left(2.5\right)}^{2}[/latex]
Step 5. Solve the equation.	[latex]A\approx \left(3.14\right){\left(2.5\right)}^{2}[/latex] [latex]A\approx 19.625\text{ sq. ft}[/latex]
Step 6. Check. Does this answer make sense?	Yes. If we draw a square around the circle, its sides would be [latex]5[/latex] ft, as shown in part 1. So the area of the square would be [latex]25[/latex] sq. ft. This is slightly more than the circle’s area, [latex]19.625[/latex] sq. ft.
Step 7. Answer the question.	The area of the circle is [latex]19.625[/latex] square feet.

A circular table has a diameter of four feet. What is the circumference of the table?

Show Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the table
Step 3. Name. Choose a variable to represent it.	Let c = the circumference of the table
Step 4. Translate. Write the appropriate formula for the situation. Substitute.	[latex]C=\pi d[/latex] [latex]C=\pi \left(4\right)[/latex]
Step 5. Solve the equation, using [latex]3.14[/latex] for [latex]\pi[/latex].	[latex]C\approx \left(3.14\right)\left(4\right)[/latex] [latex]C\approx 12.56\text{ feet}[/latex]
Step 6. Check Does this answer make sense?	If we put a square around the circle, its side would be [latex]4[/latex]. The perimeter would be [latex]16[/latex]. It makes sense that the circumference of the circle, [latex]12.56[/latex], is a little less than [latex]16[/latex].
Step 7. Answer the question.	The diameter of the table is [latex]12.56[/latex] square feet.

Find the diameter of a circle with a circumference of [latex]47.1[/latex] centimeters.

Show Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	[latex]C=47.1[/latex]cm
Step 2. Identify what you are looking for.	the diameter of the circle
Step 3. Name. Choose a variable to represent it.	Let [latex]d[/latex] = the diameter of the circle
Step 4. Translate.
Write the formula. Substitute, using [latex]3.14[/latex] to approximate [latex]\pi[/latex] .	[latex]C=\pi{d}[/latex] [latex]47.1\approx{3.14d}[/latex]
Step 5. Solve.	[latex]\Large\frac{47.1}{3.14}\normalsize\approx \Large\frac{3.14d}{3.14}[/latex] [latex]15\approx{d}[/latex]
Step 6. Check.	[latex]C=\pi{d}[/latex] [latex]47.1\stackrel{?}{=}\left(3.14\right)\left(15\right)[/latex] [latex]47.1=47.1\quad\checkmark[/latex]
Step 7. Answer the question.	The diameter of the circle is approximately [latex]15[/latex] centimeters.

Area and Circumference of a Circle when Given Fractions

Sometimes we are given the diameter or radius of a circle in fractions. Recall earlier when we were given the approximations of pi, we were told [latex]\Large\frac{22}{7}[/latex] is the fraction approximation of pi. If you use your calculator, the decimal number will fill up the display and show [latex]3.14285714[/latex]. But if we round that number to two decimal places, we get [latex]3.14[/latex], the decimal approximation of [latex]\pi[/latex]. When we have a circle with radius given as a fraction, we can substitute [latex]{\Large\frac{22}{7}}[/latex] for [latex]\pi[/latex] instead of [latex]3.14[/latex].

A circle has radius [latex]{\Large\frac{14}{15}}[/latex] meters. Approximate its:

1. Circumference
2. Area

Show Solution

1. Circumference
Find the circumference when [latex]r={\Large\frac{14}{15}}[/latex]
Write the formula for circumference.	[latex]C=2\pi \mathit{\text{r}}[/latex]
Substitute [latex]\Large\frac{22}{7}[/latex] for [latex]\pi[/latex] and [latex]\Large\frac{14}{15}[/latex] for [latex]r[/latex] .	[latex]C\approx 2\left({\Large\frac{22}{7}}\right)\left({\Large\frac{14}{15}}\right)[/latex]
Multiply.	[latex]C\approx {\Large\frac{88}{15}}\text{meters}[/latex]

2. Area
Find the area when [latex]r={\Large\frac{14}{15}}[/latex].
Write the formula for area.	[latex]A=\pi {\mathit{\text{r}}}^{2}[/latex]
Substitute [latex]\Large\frac{22}{7}[/latex] for [latex]\pi[/latex] and [latex]\Large\frac{14}{15}[/latex] for [latex]r[/latex] .	[latex]A\approx {\Large(\frac{22}{7})}{\Large(\frac{14}{15})}^{2}[/latex]
Multiply.	[latex]A\approx {\Large\frac{616}{225}}\text{square meters}[/latex]