Topics
8. Circles
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United States of AmericaVirginia
Grade 6

8.02 Circumference and pi

Lesson
Practice

Circumference and pi

A circle with a dashed line around its boundary or circumference.

The circumference of a circle is the distance around its boundary. We can think of the circumference as the perimeter of the circle.

Exploration

Use the applet to explore the relatiosnhips between parts of a circle. Drag the point on the circumference of the circle to change the diameter. Move the slider to unravel the circle.

Loading interactive...
  1. Complete the table for various values of diameter and circumference. Recall that the circumference is the length of one full rotation around the circle.
    diameter5 \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ }
    circumference15.71
  2. Try out some calculations between the diameter and circumference. Do you notice a relationship between the diameter and circumference? Explain.
  3. Do you think there could be a relationship between circumference and radius? If so, describe the relationship.

Consider this image with 3 segments the length of the diameter wrapped around the circle:

An image showing how 3 diamters of a circle is wrapped around the said circle, and 0.14159... portion is not covered.

When 3 diameters are wrapped around a circle, there is a gap of approximately 0.14159 d to complete the circle. This approximation of the circumference is equal to 3.14159 segments the length of the diameter. We use the symbol \pi pronounced "pi" to represent this number.

If the circumference is equal to 3.14159 segments the length of the diameter, we can write:

\displaystyle C=\pi d
\bm{C}
Circumference of the circle
\bm{\pi}
approximately 3.14
\bm{d}
Diameter of the circle

Now consider wrapping segments the length of the radius around the circle:

An image showing how the circumference of a circle is equal to 6 of its radius, with a remaining portion equal to 0.28318...

When 6 segments the length of the radius are wrapped around a circle, there is a gap equal to approximately 0.2832 r to complete the circle. This is an approximation of 6.28 = 2\pi segments the length of the radius. In other words:

\displaystyle C=2\pi r
\bm{C}
Circumference of the circle
\bm{\pi}
approximately 3.14
\bm{r}
Radius of the circle

Both of these formulas show us that the circumference is proportional to both the radius and diameter. In fact, we can precisely calculate \pi using this ratio:

\pi = \text{Circumference} \div \text{Diameter}

Examples

Example 1

Of these statements about \pi, which two are true?

A
\pi is exactly equal to 3.142
B
\pi cannot be expressed as a fraction using whole numbers
C
\pi is the ratio between the circumference and the radius of a circle
D
\pi is exactly equal to \dfrac{22}{7}
E
\pi is the ratio between the circumference and the diameter of a circle
F
\pi is exactly equal to 3.14
Worked Solution
Create a strategy

Recall the meaning of \pi and its ratio.

Apply the idea

Options A, D, and F are false because they are an approximation of the value of \pi but not its exact value.

Option B is true because \pi cannot be expressed as a fraction using whole numbers. It is an infinite, non-repeating decimal number that goes on forever without ending or repeating. While we can approximate the value of \pi using decimals or fractions, it can never be represented exactly by a finite number of digits.

Option C is false because \pi is the ratio between the circumference and the diameter of a circle, not the radius.

Option E is also true because the value of \pi is defined as the ratio between the circumference and the diameter of a circle. This means that if we divide the circumference of any circle by its diameter, we will always get the value of \pi. This relationship holds true for all circles, regardless of their size or shape, making \pi a fundamental constant in mathematics and geometry.

Therefore, the two correct statements are options B and E.

Example 2

Find the circumference of the circle shown, correct to two decimal places.

A circle with a radius of 6 centimeters.
Worked Solution
Create a strategy

The circumference of the circle can be found using the formula: C=2\pi r.

Apply the idea
\displaystyle C\displaystyle =\displaystyle 2\cdot \pi \cdot 6Substitute r=6
\displaystyle =\displaystyle 37.70\operatorname{ cm}Evaluate

Example 3

Find the circumference of the circle shown, correct to two decimal places.

A circle with a diameter of 13 centimeters.
Worked Solution
Create a strategy

The circumference of the circle can be found using the formula: C=\pi d.

Apply the idea
\displaystyle C\displaystyle =\displaystyle \pi \cdot 13Substitute d=13
\displaystyle =\displaystyle 40.84 \operatorname{ cm}Evaluate

Example 4

If the radius of a circle is equal to 17 \operatorname{cm} find its circumference correct to one decimal place.

Worked Solution
Create a strategy

The circumference of the circle can be found using the formula: C=2\pi r.

Apply the idea
\displaystyle C\displaystyle =\displaystyle 2\cdot \pi \cdot 17Substitute r=17
\displaystyle =\displaystyle 106.8\operatorname{ cm}Evaluate

Example 5

Lisa is cleaning the leaves out of the pool in her backyard. The pool is a circular shape and has a radius of 5 \operatorname{m}

What distance does Lisa cover if she walks all the way around the pool? Give your answer to one decimal place.

Worked Solution
Create a strategy

The distance around the outside of a circle is its circumference.

Apply the idea
\displaystyle C\displaystyle =\displaystyle 2 \pi rWrite the formula
\displaystyle =\displaystyle 2\cdot \pi \cdot 5Substitute r=5
\displaystyle =\displaystyle 31.4\operatorname{ m}Evaluate

Lisa will walk31.4 \operatorname{ m} around the pool.

Example 6

Carl is performing an experiment by spinning a metal weight around on the end of a nylon thread.

How far does the metal weight travel if it completes 40 revolutions on the end of a 0.65 \operatorname{ m} thread? Give your answer correct to one decimal place.

Worked Solution
Create a strategy

The total distance traveled by metal weight can be found using the formula:

\text{Total distance traveled}= \text{circumference}\cdot\text{number of revolutions}

The radius is equivalent to the length of thread.

Apply the idea

Find for circumference:

\displaystyle C\displaystyle =\displaystyle 2 \pi rWrite the formula for circumference
\displaystyle =\displaystyle 2 \cdot \pi \cdot 0.65Substitute r=0.65
\displaystyle =\displaystyle 1.30\pi\operatorname{ m}Evaluate
\displaystyle \text{Total distance traveled}\displaystyle =\displaystyle 1.30\pi \cdot 40Substitute circumference and number of revolutions
\displaystyle =\displaystyle 163.4\operatorname{ m}Evaluate

The metal weight traveled a total of 163.4 \operatorname{m}

Idea summary

\pi is the ratio between the circumference and diameter, which we approximate as 3.14.

The formula for circumference of a circle is :

\displaystyle C=\pi d
\bm{C}
Cirumference
\bm{d}
Diameter

and because the diameter is twice the radius, we can also write the formula as

\displaystyle C=2\pi r
\bm{C}
Circumference
\bm{r}
Radius

Outcomes

6.MG.1

The student will identify the characteristics of circles and solve problems, including those in context, involving circumference and area.

6.MG.1bii

Investigate and describe the relationship between: ii) radius and circumference;

6.MG.1biii

Investigate and describe the relationship between: iii) diameter and circumference.

6.MG.1c

Develop an approximation for pi (3.14) by gathering data and comparing the circumference to the diameter of various circles, using concrete manipulatives or technological models.

6.MG.1d

Develop the formula for circumference using the relationship between diameter, radius, and pi.

6.MG.1e

Solve problems, including those in context, involving circumference and area of a circle when given the length of the diameter or radius.

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