3.2.2: Exploring the Area of a Circle
- Page ID
- 38176
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
Lesson
Let's investigate the areas of circles.
Exercise \(\PageIndex{1}\): Estimating Areas
Your teacher will show you some figures. Decide which figure has the largest area. Be prepared to explain your reasoning.
Exercise \(\PageIndex{2}\): Estimating Areas of Circles
Your teacher will give your group two circles of different sizes.
- Set the diameter of your assigned circle and use the applet to help estimate the area of the circle.
Note: to create a polygon, select the Polygon tool, and click on each vertex. End by clicking the first vertex again. For example, to draw triangle \(ABC\), click on \(A-B-C-A\).
Figure \(\PageIndex{1}\) - Record the diameter in column \(D\) and the corresponding area in column \(A\) for your circles and others from your classmates.
- In a previous lesson, you graphed the relationship between the diameter and circumference of a circle. How is this graph the same? How is it different?
Are you ready for more?
How many circles of radius 1 unit can you fit inside each of the following so that they do not overlap?
- a circle of radius 2 units?
- a circle of radius 3 units?
- a circle of radius 4 units?
If you get stuck, consider using coins or other circular objects.
Exercise \(\PageIndex{3}\): Covering a Circle
Here is a square whose side length is the same as the radius of the circle.
How many of these squares do you think it would take to cover the circle exactly?
Summary
The circumference \(C\) of a circle is proportional to the diameter \(d\), and we can write this relationship as \(C=\pi d\). The circumference is also proportional to the radius of the circle, and the constant of proportionality is \(2\cdot\pi\) because the diameter is twice as long as the radius. However, the area of a circle is not proportional to the diameter (or the radius).
The area of a circle with radius \(r\) is a little more than 3 times the area of a square with side \(r\) so the area of a circle of radius \(r\) is approximately \(3r^{2}\). We saw earlier that the circumference of a circle of radius \(r\) is \(2\pi r\). If we write \(C\) for the circumference of a circle, this proportional relationship can be written \(C=2\pi r\).
The area \(A\) of a circle with radius \(r\) is approximately \(3r^{2}\). Unlike the circumference, the area is not proportional to the radius because \(3r^{2}\) cannot be written in the form \(kr\) for a number \(k\). We will investigate and refine the relationship between the area and the radius of a circle in future lessons.
Glossary Entries
Definition: Area of a Circle
If the radius of a circle is \(r\) units, then the area of the circle is \(\pi r^{2}\) square units.
For example, a circle has radius 3 inches. Its area is \(\pi 3^{2}\) square inches, or \(9\pi\) square inches, which is approximately 28.3 square inches.
Practice
Exercise \(\PageIndex{4}\)
The \(x\)-axis of each graph has the diameter of a circle in meters. Label the \(y\)-axis on each graph with the appropriate measurement of a circle:
radius (m), circumference (m), or area (m2).
Exercise \(\PageIndex{5}\)
Circle A has area 500 in2. The diameter of circle B is three times the diameter of circle A. Estimate the area of circle B.
Exercise \(\PageIndex{6}\)
Lin’s bike travels 100 meters when her wheels rotate 55 times. What is the circumference of her wheels?
(From Unit 3.1.5)
Exercise \(\PageIndex{7}\)
Priya drew a circle whose circumference is 25 cm. Clare drew a circle whose diameter is 3 times the diameter of Priya’s circle. What is the circumference of Clare’s circle?
(From Unit 3.1.3)
Exercise \(\PageIndex{8}\)
1. Here is a picture of two squares and a circle. Use the picture to explain why the area of this circle is more than 2 square units but less than 4 square units.
2. Here is another picture of two squares and a circle. Use the picture to explain why the area of this circle is more than 18 square units and less than 36 square units.
Exercise \(\PageIndex{9}\)
Point \(A\) is the center of the circle, and the length of \(CD\) is 15 centimeters. Find the circumference of this circle.
(From Unit 3.1.3)