Question

worksheet: circumference of a circle

1. what is the formula for the circumference of a circle in terms of its radius?
2. a circle has a radius of 5 cm. what is its circumference? write your answer in terms of $pi$.
3. if the diameter of a circle is 12 inches, what is the circumference of the circle in terms of $pi$?
4. explain how the circumference of a circle changes if its radius is doubled.
5. a circular swimming pool has a circumference of 31.4 meters. what is the radius of the pool? use $pi \approx 3.14$.
6. find the circumference of a circle with a diameter of 20 cm. use $pi = 3.14$.
7. if the circumference of a circle is $18\pi$ feet, what is its diameter?
8. describe the relationship between the diameter of a circle and its circumference.

Explanation:

Step1: Recall radius-based circumference formula

The circumference formula using radius $r$ is $C = 2\pi r$.

Step2: Calculate circumference for $r=5$ cm

Substitute $r=5$ into $C=2\pi r$:
$C = 2\pi \times 5 = 10\pi$

Step3: Relate diameter to circumference

Since diameter $d=2r$, $C=\pi d$. Substitute $d=12$:
$C = \pi \times 12 = 12\pi$

Step4: Analyze radius doubling effect

Let original radius be $r$, new radius $2r$. Original $C_1=2\pi r$, new $C_2=2\pi(2r)=4\pi r=2C_1$.

Step5: Solve for radius from circumference

Rearrange $C=2\pi r$ to $r=\frac{C}{2\pi}$. Substitute $C=31.4$, $\pi\approx3.14$:
$r = \frac{31.4}{2\times3.14} = \frac{31.4}{6.28} = 5$

Step6: Calculate circumference for $d=20$ cm

Use $C=\pi d$, substitute $d=20$, $\pi=3.14$:
$C = 3.14 \times 20 = 62.8$

Step7: Solve for diameter from $C=18\pi$

Rearrange $C=\pi d$ to $d=\frac{C}{\pi}$. Substitute $C=18\pi$:
$d = \frac{18\pi}{\pi} = 18$

Step8: Define diameter-circumference relationship

Circumference is $\pi$ times the diameter, a proportional relationship.

Answer:

1. $C = 2\pi r$
2. $10\pi$ cm
3. $12\pi$ inches
4. The circumference doubles.
5. 5 meters
6. 62.8 cm
7. 18 feet
8. The circumference of a circle is equal to $\pi$ multiplied by its diameter, so they are directly proportional.