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Studienführer > Prealgebra

Applications with Pi

Learning Outcomes

  • Find the circumference of a circle
  • Find the area of a circle
  The properties of circles have been studied for over 2,0002,000 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. See the image below. The size of a circle can be measured in two ways. The distance around a circle is called its circumference. A circle is shown. A dotted line running through the widest portion of the circle is labeled as a diameter. A dotted line from the center of the circle to a point on the circle is labeled as a radius. Along the edge of the circle is the 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 π\pi (pronounced "pie"). However, the exact value of π\pi cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers.) Doing the Manipulative Mathematics activity Pi Lab will help you develop a better understanding of pi. If we want the exact circumference or area of a circle, we leave the symbol π\pi in the answer. We can get an approximate answer by substituting 3.143.14 as the value of π\pi . We use the symbol ≈\approx to show that the result is approximate, not exact.

Properties of Circles

A circle is shown. A line runs through the widest portion of the circle. There is a red dot at the center of the circle. The half of the line from the center of the circle to a point on the right of the circle is labeled with an r. The half of the line from the center of the circle to a point on the left of the circle is also labeled with an r. The two sections labeled r have a brace drawn underneath showing that the entire segment is labeled d. r is the length of the radius.d is the length of the diameter.\begin{array}{c}r\text{ is the length of the radius.}\hfill \\ d\text{ is the length of the diameter.}\hfill \end{array} The circumference is 2πr.C=2πrThe area is πr2.A=πr2\begin{array}{cccc}\text{The circumference is }2\pi \mathit{\text{r}}.\hfill & & & C=2\pi \mathit{\text{r}}\hfill \\ \text{The area is }\pi{\mathit{\text{r}}}^{2}.\hfill & & & A=\pi{\mathit{\text{r}}}^{2}\hfill \end{array}
  Since the diameter is twice the radius, another way to find the circumference is to use the formula C=πdC=\pi \mathit{\text{d}}. Suppose we want to find the exact area of a circle of radius 1010 inches. To calculate the area, we would evaluate the formula for the area when r=10r=10 inches and leave the answer in terms of π\pi. A=πr2A=π(102)A=π⋅100\begin{array}{}\\ A=\pi {\mathit{\text{r}}}^{2}\hfill \\ A=\pi \text{(}{10}^{2}\text{)}\hfill \\ A=\pi \cdot 100\hfill \end{array} We write π\pi after the 100100. So the exact value of the area is A=100πA=100\pi square inches. To approximate the area, we would substitute π≈3.14\pi \approx 3.14. A=100π≈100⋅3.14≈314 square inches\begin{array}{ccc}A& =& 100\pi \hfill \\ \\ & \approx & 100\cdot 3.14\hfill \\ & \approx & 314\text{ square inches}\hfill \end{array} Remember to use square units, such as square inches, when you calculate the area.  

example

A circle has radius 1010 centimeters. Approximate its ⓐ circumference and ⓑ area. Solution
ⓐ Find the circumference when r=10r=10.
Write the formula for circumference. C=2πrC=2\pi \mathit{\text{r}}
Substitute 3.143.14 for π\pi and 10 for , rr . C≈2(3.14)(10)C\approx 2\left(3.14\right)\left(10\right)
Multiply. C≈62.8 centimetersC\approx 62.8\text{ centimeters}
ⓑ Find the area when r=10r=10.
Write the formula for area. A=πr2A=\pi {\mathit{\text{r}}}^{2}
Substitute 3.143.14 for π\pi and 10 for rr . A≈(3.14)(10)2A\approx \left(3.14\right){\text{(}10\text{)}}^{2}
Multiply. A≈314 square centimetersA\approx 314\text{ square centimeters}
 

try it

[ohm_question]146563[/ohm_question]
   

example

A circle has radius 42.542.5 centimeters. Approximate its ⓐ circumference and ⓑ area.

Answer: Solution

ⓐ Find the circumference when rr=42.542.5.
Write the formula for circumference. C=2πrC=2\pi \mathit{\text{r}}
Substitute 3.143.14 for π\pi and 42.542.5 for rr C≈2(3.14)(42.5)C\approx 2\left(3.14\right)\left(42.5\right)
Multiply. C≈266.9 centimetersC\approx 266.9\text{ centimeters}
ⓑ Find the area when r=42.5r=42.5 .
Write the formula for area. A=πr2A=\pi {\mathit{\text{r}}}^{2}
Substitute 3.143.14 for π\pi and 42.542.5 for rr . A≈(3.14)(42.5)2A\approx \left(3.14\right){\text{(}42.5\text{)}}^{2}
Multiply. A≈5671.625 square centimetersA\approx 5671.625\text{ square centimeters}

 

try it

[ohm_question]146564[/ohm_question]
Watch the following video to see another example of how to find the circumference of a circle. https://youtu.be/sHtsnC2Mgnk In the next video example, we find the area of a circle. https://youtu.be/SIKkWLqt2mQ

Approximate π\pi with a Fraction

Convert the fraction 227\frac{22}{7} to a decimal. If you use your calculator, the decimal number will fill up the display and show 3.142857143.14285714. But if we round that number to two decimal places, we get 3.143.14, the decimal approximation of π\pi . When we have a circle with radius given as a fraction, we can substitute 227\frac{22}{7} for π\pi instead of 3.143.14. And, since 227\frac{22}{7} is also an approximation of π\pi , we will use the ≈\approx symbol to show we have an approximate value.  

example

A circle has radius 1415\frac{14}{15} meter. Approximate its ⓐ circumference and ⓑ area.

Answer: Solution

ⓐ Find the circumference when r=1415r=\frac{14}{15}.
Write the formula for circumference. C=2πrC=2\pi \mathit{\text{r}}
Substitute 227\frac{22}{7} for π\pi and 1415\frac{14}{15} for rr . C≈2(227)(1415)C\approx 2\left(\frac{22}{7}\right)\left(\frac{14}{15}\right)
Multiply. C≈8815metersC\approx \frac{88}{15}\text{meters}
ⓑ Find the area when r=1415r=\frac{14}{15}.
Write the formula for area. A=πr2A=\pi {\mathit{\text{r}}}^{2}
Substitute 227\frac{22}{7} for π\pi and 1415\frac{14}{15} for rr . A≈(227)(1415)2A\approx \left(\frac{22}{7}\right){\text{(}\frac{14}{15}\text{)}}^{2}
Multiply. A≈616225square metersA\approx \frac{616}{225}\text{square meters}

 

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[ohm_question]146611[/ohm_question]
 

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