The circumference of a circle
The circumference of a circle
- Unit 1: Angles & Constructions
- Unit 2: Shapes & Symmetry
- Unit 3: Position & Transformation
- Unit 4: Area, Volume & Symmetry
🎯 In this topic you will
- Know and use the formulae for the circumference and area of a circle
- Know and use the formula for the circumference of a circle
🧠 Key Words
- accurate
- approximate value
- circumference
- diameter
- pi (π)
- radius
- semicircle
Show Definitions
- accurate: Correct in all details; exact and precise without error.
- approximate value: A value that is close to but not exactly equal to the actual measurement.
- circumference: The distance around the boundary of a circle.
- diameter: A straight line passing through the center of a circle, connecting two points on the circumference.
- pi (π): The constant ratio of a circle's circumference to its diameter, approximately equal to $3.14159$.
- radius: A straight line from the center of a circle to any point on its circumference.
- semicircle: Half of a circle, formed by cutting along a diameter.
You already know the names of the parts of a circle.
Did you know there is a link between the circumference of a circle and the diameter of a circle?
This table shows the circumference and diameter measurements of four circles.
| Circle | Circumference (cm) | Diameter (cm) | Circumference ÷ diameter |
|---|---|---|---|
| A | $9.1$ | $2.9$ | |
| B | $19.8$ | $6.3$ | |
| C | $25.1$ | $8$ | |
| D | $37.1$ | $11.8$ |
Copy the table and fill in the final column. Give your answers correct to two decimal places. What do you notice?
You should notice that all the answers are $3.14$ correct to 2 decimal places.
This means that the ratio of the diameter to the circumference of a circle is approximately $1:3.14$.
The number $3.14\ldots$ has a special name, pi. It is written using the symbol $\\pi$.
$\\pi$ is the number $3.141\ 592\ 653\ 589\ldots$, but you will often use $3.14$ or $3.142$ as an approximate value for $\\pi$.
You now know that $$\frac{\text{circumference}}{\text{diameter}} = \\pi$$, so you can rearrange the formula to get:
$C = \\pi d$ where:
$C$ is the circumference of the circle
$d$ is the diameter of the circle
❓ EXERCISES
1. Copy and complete the workings to find the circumference of each circle.
Use $\pi = 3.14$. Round your answers correct to 1 decimal place (1 d.p.).
a. diameter = $6 \, cm$
$C = \pi d$
$= 3.14 \times 6$
$= \; ?$
$= \; ? \, cm$ (1 d.p.)
b. diameter = $25 \, cm$
$C = \pi d$
$= 3.14 \times 25$
$= \; ?$
$= \; ? \, cm$ (1 d.p.)
c. diameter = $4.25 \, m$
$C = \pi d$
$= 3.14 \times 4.25$
$= \; ?$
$= \; ? \, m$ (1 d.p.)
👀 Show answer
a. $3.14 \times 6 = 18.84 \approx 18.8 \, cm$
b. $3.14 \times 25 = 78.5 \, cm$
c. $3.14 \times 4.25 = 13.345 \approx 13.3 \, m$
2. Copy and complete the workings to find the circumference of each circle.
Use $\pi = 3.142$. Round your answers correct to 2 decimal places (2 d.p.).
a. radius = $7 \, cm$
$d = 2 \times r$
$= 2 \times 7$
$= 14 \, cm$
$C = \pi d$
$= 3.142 \times 14$
$= \; ?$
$= \; ? \, cm$ (2 d.p.)
b. radius = $2.6 \, cm$
$d = 2 \times r$
$= 2 \times 2.6$
$= 5.2 \, cm$
$C = \pi d$
$= 3.142 \times 5.2$
$= \; ?$
$= \; ? \, cm$ (2 d.p.)
c. radius = $0.9 \, m$
$d = 2 \times r$
$= 2 \times 0.9$
$= 1.8 \, m$
$C = \pi d$
$= 3.142 \times 1.8$
$= \; ?$
$= \; ? \, m$ (2 d.p.)
👀 Show answer
a. $3.142 \times 14 = 43.988 \approx 43.99 \, cm$
b. $3.142 \times 5.2 = 16.3384 \approx 16.34 \, cm$
c. $3.142 \times 1.8 = 5.6556 \approx 5.66 \, m$
🧠 Think like a Mathematician
So far in this unit you have used approximate values for $\pi$.
You have used $\pi = 3.14$ and $\pi = 3.142$.
There is another approximate value you can use: $\pi = \dfrac{22}{7}$.
A more accurate value for $\pi$ is stored on your calculator. Can you find the button with the $\pi$ symbol on it?
a. Use the $\pi$ button on your calculator to work out the accurate circumference of a circle with diameter $12\,\text{cm}$.
Write all the numbers on your calculator screen.
b. Now work out the circumference of the same circle using approximate values for $\pi$ of:
i.$3.14$ ii.$3.142$ iii.$\dfrac{22}{7}$
c. Compare your answers to parts a and b.
Which approximate value for $\pi$ gives the closest answer to the accurate answer?
d. When you answer questions and you need to use $\pi$, which value of $\pi$ do you think it is best to use?
Explain why.
👀 Show Answer
a. Use the calculator value for $\pi$ (e.g., $3.141592654$).
$C = \pi d = 12\pi \approx 12 \times 3.141592654 = 37.69911184\,\text{cm}$.
To two d.p., $37.70\,\text{cm}$. (Your screen will show numbers like $3.141592654\times 12 = 37.69911184$.)
b.
i.$C = 3.14 \times 12 = 37.68\,\text{cm}$
ii.$C = 3.142 \times 12 = 37.704\,\text{cm}$
iii.$C = \dfrac{22}{7} \times 12 = 37.\overline{714285}\,\text{cm}$
c. The accurate value is $37.699\ldots$.
Differences: $|37.699-37.68| \approx 0.019$, $|37.699-37.704| \approx 0.005$, $|37.699-37.714| \approx 0.015$.
So $3.142$ is the closest approximation among the three.
d. Best: use the $\pi$ button for maximum accuracy. If you must use a fixed decimal, choose $3.142$ because it is closer to the true value than $3.14$ and usually closer than $\dfrac{22}{7}$ for typical rounding.
❓ EXERCISES
4. Work out the circumference of each circle. Use the $\pi$ button on your calculator.
Round your answers correct to 2 decimal places (2 d.p.).
a. diameter = $9 \, cm$
b. diameter = $7.25 \, m$
c. radius = $11 \, cm$
d. radius = $3.2 \, m$
👀 Show answer
a. $C = \pi d = 9\pi \approx 28.2743 \; cm \approx 28.27 \; cm$
b. $C = \pi d = 7.25\pi \approx 22.7765 \; m \approx 22.78 \; m$
c. $d = 2r = 22 \; cm$, $C = 22\pi \approx 69.1150 \; cm \approx 69.12 \; cm$
d. $d = 2r = 6.4 \; m$, $C = 6.4\pi \approx 20.1062 \; m \approx 20.11 \; m$
🧠 Think like a Mathematician
So far in this unit you have used the formula $C = \pi d$.
In questions 2 and 4, you found the circumference when you were given the radius.
Task: Can you write a formula to find the circumference which uses $r$ (radius) instead of $d$ (diameter)?
Test your formula on Question 4, parts c and d. Does it work?
Compare your formula with other approaches to check consistency.
👀 Show Answer
The diameter is related to the radius by $d = 2r$. Substituting into $C = \pi d$ gives:
$C = \pi (2r) = 2\pi r$
Check with Question 4:
- c. $r = 11 \, cm$, $C = 2\pi r = 2\pi \times 11 = 22\pi \approx 69.12 \, cm$ ✅
- d. $r = 3.2 \, m$, $C = 2\pi r = 2\pi \times 3.2 = 6.4\pi \approx 20.11 \, m$ ✅
Yes, the formula works consistently.
❓ EXERCISES
6. Fu and Fern use different methods to work out the answer to this question.
Work out the diameter of a circle with circumference $16.28 \text{ cm}$.
Give your answer correct to $3$ significant figures.
This is what they write:
a. Look at Fu and Fern’s methods.
Do you understand both methods?
Do you think you would be able to use both methods?
👀 Show answer
b. Which method do you prefer and why?
👀 Show answer
c. Use your preferred method to work out the diameter of a circle with:
i. circumference $= 28 \text{ cm}$
ii. circumference $= 4.58 \text{ m}$
👀 Show answer
ii. $d = \dfrac{4.58}{\pi} \approx 1.46 \text{ m}$
d. Make $r$ the subject of the formula $C = 2\pi r$
👀 Show answer
e. Use your formula from part d to work out the radius of a circle with:
i. circumference $= 15 \text{ cm}$
ii. circumference $= 9.25 \text{ m}$
👀 Show answer
ii. $r = \dfrac{9.25}{2\pi} \approx 1.47 \text{ m}$
7. The circumference of a circular disc is $39 \text{ cm}$.
Work out the diameter of the disc.
Give your answer correct to the nearest millimetre.
👀 Show answer
❓ EXERCISES
8. A circular ring has a circumference of $5.65\,cm$.
Show that the radius of the ring is $9\,mm$, correct to the nearest millimetre.
👀 Show answer
Use $C = 2\pi r$. Then $r = \dfrac{C}{2\pi} = \dfrac{5.65}{2\pi}\,cm \approx 0.899\,cm$.
Convert to millimetres: $0.899\,cm = 8.99\,mm$, which rounds to $9\,mm$ (nearest mm).
9. This is part of Ahmad’s homework.
a. Use Ahmad’s method to work out the perimeter of a semicircle with:
i. diameter = $20\,cm$ ii. diameter = $15\,m$
iii. radius = $8\,cm$ iv. radius = $6.5\,m$
Round your answers correct to $2$ d.p.
b. Imagine you have a friend who does not know how to work out the perimeter of a semicircle.
Are you confident you could explain to them how to work it out?
Can you use your knowledge to explain how to work out the perimeter of a quarter-circle?
Make a sketch of a quarter-circle to help you.
👀 Show answer
Method (Ahmad): For a semicircle with diameter $d$, perimeter $P = \dfrac{\pi d}{2} + d$. If radius $r$ is given, use $d=2r$.
a. Answers
- i.$P = \dfrac{\pi\times 20}{2} + 20 = 10\pi + 20 \approx 51.42\,cm$
- ii.$P = \dfrac{\pi\times 15}{2} + 15 = \dfrac{15\pi}{2} + 15 \approx 38.56\,m$
- iii.$d=2r=16\,cm$, so $P = \dfrac{\pi\times 16}{2} + 16 = 8\pi + 16 \approx 41.13\,cm$
- iv.$d=2r=13\,m$, so $P = \dfrac{\pi\times 13}{2} + 13 = \dfrac{13\pi}{2} + 13 \approx 33.42\,m$
b. Explanation: A semicircle’s curved edge is half a circle’s circumference $(\tfrac{1}{2}\times 2\pi r=\pi r)$, then add the diameter $(2r)$. So $P_{\tfrac12\text{circle}} = \pi r + 2r = r(\pi + 2)$ or in terms of $d$, $\dfrac{\pi d}{2} + d$.
For a quarter-circle, use a quarter of the circumference plus two radii: $P_{\tfrac14\text{circle}} = \tfrac{1}{4}(2\pi r) + 2r = \dfrac{\pi r}{2} + 2r$.
10. The diagram shows a semicircle and a quarter-circle. Read what Zara says.
“I think the perimeter of the semicircle is greater than the perimeter of the quarter-circle.”
Is Zara correct? Show working to support your answer.
👀 Show answer
Semicircle: diameter $15\,m$ → $P_s = \dfrac{\pi d}{2} + d = \dfrac{\pi\times 15}{2} + 15 \approx 38.56\,m$.
Quarter-circle: radius $10\,m$ → $P_q = \dfrac{\pi r}{2} + 2r = \dfrac{\pi\times 10}{2} + 2\times 10 = 5\pi + 20 \approx 35.71\,m$.
Since $38.56\,m > 35.71\,m$, Zara is correct: the semicircle has the greater perimeter.
❓ EXERCISES
11. Work out the perimeter of each compound shape. Give your answers correct to two decimal places.
👀 Show answer
a. Triangle sides on the outside: $14.4 + 8$; plus a semicircle of diameter $12$: $\tfrac{\pi d}{2} = 6\pi$.
$P = 14.4 + 8 + 6\pi \approx 41.25\,cm$.
b. Stadium with a $4.5\,m \times 4.5\,m$ rectangle and one semicircular end of diameter $4.5\,m$.
Straight sides: $4.5 + 4.5 + 4.5$; arc: $\tfrac{\pi d}{2} = 2.25\pi$.
$P = 13.5 + 2.25\pi \approx 20.57\,m$.
c. Top large semicircle, $d=56\,mm$ (because $28\,mm+28\,mm$); bottom small semicircle, $d=28\,mm$; straight base parts sum to $28+28=56\,mm$.
Arcs: $\tfrac{\pi\times 56}{2} + \tfrac{\pi\times 28}{2} = 28\pi + 14\pi = 42\pi$.
$P = 56 + 42\pi \approx 187.95\,mm$.
d. Two semicircles each of diameter $3.6\,cm$ plus two equal slanted sides $4.5\,cm$.
Arcs: $2\times \tfrac{\pi d}{2} = 2\times 1.8\pi = 3.6\pi$; straights: $4.5+4.5=9$.
$P = 9 + 3.6\pi \approx 20.31\,cm$.
🟣 Circumference Formula
You already know how to work out the circumference of a circle using this formula:
$C=\pi d$ where: $C$ is the circumference of the circle
$d$ is the diameter of the circle
🟡 Area Formula
You can work out the area of a circle using this formula:
$A=\pi r^2$ where: $A$ is the area of the circle
$r$ is the radius of the circle
🔎 Comparing Formulas
Notice that the formula for the circumference uses the diameter while the formula for the area uses the radius.
❓ EXERCISES
1. Copy and complete the workings to work out the area of each circle.
Use $\pi = 3.14$. Round your answers correct to $1$ decimal place (1 d.p.).
a. radius = $2\,cm$
$A = \pi r^2$
$= 3.14 \times 2^2$
$= 3.14 \times\; ?$
$= \; ?\; cm^2$ (1 d.p.)
b. radius = $9\,cm$
$A = \pi r^2$
$= 3.14 \times 9^2$
$= 3.14 \times\; ?$
$= \; ?\; cm^2$ (1 d.p.)
c. radius = $4.2\,m$
$A = \pi r^2$
$= 3.14 \times 4.2^2$
$= 3.14 \times\; ?$
$= \; ?\; m^2$ (1 d.p.)
👀 Show answer
a. $2^2=4$, so $A=3.14\times 4=12.56\,cm^2\approx \mathbf{12.6\,cm^2}$.
b. $9^2=81$, so $A=3.14\times 81=254.34\,cm^2\approx \mathbf{254.3\,cm^2}$.
c. $4.2^2=17.64$, so $A=3.14\times 17.64=55.3896\,m^2\approx \mathbf{55.4\,m^2}$.
2. Copy and complete the workings to work out the area of each circle.
Use $\pi = 3.142$. Round your answers correct to $2$ decimal places (2 d.p.).
a. diameter = $16\,cm$
$r = d \div 2$
$= 16 \div 2$
$= 8\,cm$
$A = \pi r^2$
$= 3.142 \times 8^2$
$= 3.142 \times\; ?$
$= \; ?\; cm^2$ (2 d.p.)
b. diameter = $9\,cm$
$r = d \div 2$
$= 9 \div 2$
$= 4.5\,cm$
$A = \pi r^2$
$= 3.142 \times 4.5^2$
$= 3.142 \times\; ?$
$= \; ?\; cm^2$ (2 d.p.)
c. diameter = $2.6\,m$
$r = d \div 2$
$= 2.6 \div 2$
$= 1.3\,m$
$A = \pi r^2$
$= 3.142 \times 1.3^2$
$= 3.142 \times\; ?$
$= \; ?\; m^2$ (2 d.p.)
👀 Show answer
a. $8^2=64$, so $A=3.142\times 64=201.088\,cm^2\approx \mathbf{201.09\,cm^2}$.
b. $4.5^2=20.25$, so $A=3.142\times 20.25=63.6255\,cm^2\approx \mathbf{63.63\,cm^2}$.
c. $1.3^2=1.69$, so $A=3.142\times 1.69=5.30998\,m^2\approx \mathbf{5.31\,m^2}$.
❓ EXERCISES
3. Work with a partner or individually to answer this question.
a. Use the $\pi$ button on your calculator to work out the accurate area of a circle with radius $7\,cm$.
Write your answer correct to three decimal places (3 d.p.).
b. Now work out the area of the same circle using approximate values for $\pi$ of:
i.$3.14$ ii.$3.142$ iii.$\dfrac{22}{7}$
Tip
In percentage difference, use a positive difference: accurate minus approximate or approximate minus accurate — choose the one that gives a positive value.
c. Use this formula to work out the percentage difference between the accurate answer (part a) and each of the approximate answers in part b. Give your percentages correct to two decimal places (2 d.p.).
$\text{percentage difference} = \dfrac{\text{difference between accurate and approximate answers}}{\text{accurate answer}} \times 100$
d. Which approximate value for $\pi$ gives the smallest percentage difference?
e. When you answer questions and you need to use $\pi$, which value of $\pi$ do you think it is best to use? Explain why.
👀 Show answer
a. $A = \pi r^2 = \pi \times 7^2 = 49\pi \approx \mathbf{153.938\,cm^2}$ (3 d.p.).
b.
i. $A \approx 3.14 \times 49 = \mathbf{153.86\,cm^2}$
ii. $A \approx 3.142 \times 49 = \mathbf{153.958\,cm^2}$
iii. $A \approx \dfrac{22}{7} \times 49 = \mathbf{154.00\,cm^2}$
c. Using $\dfrac{|\text{approx} - \text{accurate}|}{\text{accurate}}\times 100$:
i. $\dfrac{|153.86 - 153.938|}{153.938}\times 100 \approx \mathbf{0.05\%}$
ii. $\dfrac{|153.958 - 153.938|}{153.938}\times 100 \approx \mathbf{0.01\%}$
iii. $\dfrac{|154.00 - 153.938|}{153.938}\times 100 \approx \mathbf{0.04\%}$
d. The smallest percentage difference is with $3.142$.
e. Use the $\pi$ button for maximum accuracy. If a fixed decimal is required, choose $3.142$ because it is closer to the true value of $\pi$ than $3.14$ and typically closer than $\dfrac{22}{7}$ for rounding to a few d.p.
❓ EXERCISES
4. Work out the area of each circle. Use the $\pi$ button on your calculator.
Round your answers correct to $3$ significant figures ($3$ s.f.).
a. radius = $6\,cm$
b. radius = $4.25\,m$
c. diameter = $23\,cm$
d. diameter = $4.8\,m$
👀 Show answer
a. $A = \pi r^2 = \pi\times 6^2 = 36\pi \approx \mathbf{113\,cm^2}$ (3 s.f.).
b. $A = \pi r^2 = \pi\times 4.25^2 = \pi\times 18.0625 \approx \mathbf{56.7\,m^2}$ (3 s.f.).
c. $r=\tfrac{d}{2}=\tfrac{23}{2}=11.5\,cm$, so $A=\pi r^2=\pi\times 11.5^2=\pi\times 132.25 \approx \mathbf{415\,cm^2}$ (3 s.f.).
d. $r=\tfrac{d}{2}=\tfrac{4.8}{2}=2.4\,m$, so $A=\pi r^2=\pi\times 2.4^2=\pi\times 5.76 \approx \mathbf{18.1\,m^2}$ (3 s.f.).
5. Ellie and Hans work out the answer to this question.
Question
Work out the area of a circle with radius $1.7\,m$. Use $\pi = 3.14$.
This is what they write:
Ellie
$3.14 \times 1.7 = 5.338$
$5.338^2 = 28.449244$
Area $= 28.5\,m^2\; (3\;\text{s.f.})$
Hans
$1.7^2 = 3.4$
$3.14 \times 3.4 = 10.676$
Area $= 10.7\,m^2\; (3\;\text{s.f.})$
a. Critique their solutions. Explain any mistakes they have made.
b. Write a full worked solution to show the correct way of answering the question.
👀 Show answer
a. Ellie multiplied $\pi$ by $r$ then squared the result. The formula is $A = \pi r^2$, so only the radius is squared: square $r$ first, then multiply by $\pi$. Hans computed $1.7^2$ incorrectly; it is $2.89$, not $3.4$.
b. Correct method with $\pi = 3.14$:
$A = \pi r^2 = 3.14 \times (1.7)^2 = 3.14 \times 2.89 = 9.0746\,m^2 \approx \mathbf{9.07\,m^2}$ (3 s.f.).
🧠 Think like a Mathematician
Task: Work on this problem independently.
So far in this unit you have used the formula $A = \pi r^2$.
- In questions 1, 4a and 4b, you found the area when given the radius.
- In questions 2, 4c and 4d, you found the area when given the diameter.
Can you write a formula to work out the area which uses $d$ (diameter) instead of $r$ (radius)?
Write your formula in its simplest form. Then test it on questions 4c and 4d to check if it works.
Finally, compare your formula with the expected result.
👀 show answer
- a) Since $r = \tfrac{d}{2}$, substitute into $A = \pi r^2$ to get: $A = \pi \left(\tfrac{d}{2}\right)^2 = \tfrac{\pi d^2}{4}$.
- b) • For $d = 23\ \text{cm}$: $A = \tfrac{\pi \times 23^2}{4} \approx 415.5\ \text{cm}^2$. • For $d = 4.8\ \text{m}$: $A = \tfrac{\pi \times 4.8^2}{4} \approx 18.1\ \text{m}^2$.
- c) Yes, both methods give the same result because the formulas are equivalent — one uses radius, the other diameter.
❓ EXERCISES
7. Work out i the area and ii the circumference of each circle. Use the $\pi$ button on your calculator.
Give your answers correct to one decimal place (1 d.p.).
a. radius = $5.6\,\text{cm}$
b. diameter = $32\,\text{mm}$
👀 Show answer
a.
Area:$A=\pi r^2=\pi\times(5.6)^2=\pi\times31.36\approx 98.5\,\text{cm}^2$ (1 d.p.)
Circumference:$C=2\pi r=2\pi\times5.6\approx 35.2\,\text{cm}$ (1 d.p.)
b.
Here $d=32\,\text{mm}$, so $r=16\,\text{mm}$.
Area:$A=\pi r^2=\pi\times 16^2=\pi\times256\approx 804.2\,\text{mm}^2$ (1 d.p.)
Circumference:$C=\pi d=\pi\times32\approx 100.5\,\text{mm}$ (1 d.p.)
8. This is part of Pria’s homework.
Use Pria’s method to work out an estimate of the area and calculate the accurate area of a semicircle with
a. radius = $6.2\,\text{cm}$ b. radius = $14.85\,\text{m}$
c. diameter = $14.7\,\text{cm}$ d. diameter = $19.28\,\text{m}$
Round your answers correct to 2 d.p.
👀 Show answer
Method (Pria’s estimate): round the diameter (or radius) to a convenient whole number and use $\pi\approx 3$.
Accurate area of a semicircle:$A_{\tfrac12}=\tfrac{1}{2}\pi r^2=\tfrac{\pi d^2}{8}$.
a.r = $6.2\,\text{cm}$ → estimate with $r\approx 6$: $A_{\text{est}}\approx \tfrac{1}{2}\times 3\times 6^2=54.00\,\text{cm}^2$.
Accurate: $A_{\text{acc}}=\tfrac{1}{2}\pi(6.2)^2\approx \mathbf{60.38\,\text{cm}^2}$ (2 d.p.).
b.r = $14.85\,\text{m}$ → estimate with $r\approx 15$: $A_{\text{est}}\approx \tfrac{1}{2}\times 3\times 15^2=337.50\,\text{m}^2$.
Accurate: $A_{\text{acc}}=\tfrac{1}{2}\pi(14.85)^2\approx \mathbf{346.40\,\text{m}^2}$ (2 d.p.).
c.d = $14.7\,\text{cm}$ → estimate with $d\approx 15$ ($r\approx 7.5$): $A_{\text{est}}\approx \tfrac{1}{2}\times 3\times 7.5^2=84.38\,\text{cm}^2$.
Accurate: $A_{\text{acc}}=\tfrac{\pi d^2}{8}=\tfrac{\pi(14.7)^2}{8}\approx \mathbf{84.86\,\text{cm}^2}$ (2 d.p.).
d.d = $19.28\,\text{m}$ → estimate with $d\approx 19$ ($r\approx 9.5$): $A_{\text{est}}\approx \tfrac{1}{2}\times 3\times 9.5^2=135.38\,\text{m}^2$.
Accurate: $A_{\text{acc}}=\tfrac{\pi d^2}{8}=\tfrac{\pi(19.28)^2}{8}\approx \mathbf{145.97\,\text{m}^2}$ (2 d.p.).
❓ EXERCISES
💡 Tip
Remember, the perimeter of a semicircle is half the circumference plus the diameter.
9. Work out i the area and ii the perimeter of each semicircle.
Give your answers correct to one decimal place (1 d.p.).
a. radius = $12.5 \, \text{m}$
👀 Show answer
i. Area
$A = \tfrac{1}{2} \pi r^2 = \tfrac{1}{2} \times \pi \times (12.5)^2$
$= \tfrac{1}{2} \times 3.1416 \times 156.25 = 245.4 \, \text{m}^2 \, (1 \, \text{d.p.})$
ii. Perimeter
$P = \tfrac{1}{2}(2 \pi r) + d = \pi r + 2r$
$= \pi \times 12.5 + 25 = 39.27 + 25 = 64.3 \, \text{m} \, (1 \, \text{d.p.})$
b. diameter = $46 \, \text{mm}$
👀 Show answer
Radius $r = \tfrac{46}{2} = 23 \, \text{mm}$
i. Area
$A = \tfrac{1}{2} \pi r^2 = \tfrac{1}{2} \times \pi \times (23)^2$
$= \tfrac{1}{2} \times 3.1416 \times 529 = 830.6 \, \text{mm}^2 \, (1 \, \text{d.p.})$
ii. Perimeter
$P = \pi r + d = \pi \times 23 + 46$
$= 72.26 + 46 = 118.3 \, \text{mm} \, (1 \, \text{d.p.})$
❓ EXERCISES
10. The diagram shows a semicircle and a quarter of a circle. Marcus makes this conjecture.
I think the area of the semicircle is greater than the area of the quarter-circle.
Is Marcus correct? Show working to support your answer.
👀 Show answer
Step 1: Semicircle area
Diameter of semicircle $= 5.2 \text{ cm}$, so radius $r = \dfrac{5.2}{2} = 2.6 \text{ cm}$.
Area of full circle $= \pi r^2 = \pi (2.6^2) = \pi (6.76) \approx 21.24 \text{ cm}^2$.
Area of semicircle $= \dfrac{1}{2} \times 21.24 \approx 10.62 \text{ cm}^2$.
Step 2: Quarter-circle area
Radius of quarter-circle $= 3.4 \text{ cm}$.
Area of full circle $= \pi r^2 = \pi (3.4^2) = \pi (11.56) \approx 36.32 \text{ cm}^2$.
Area of quarter-circle $= \dfrac{1}{4} \times 36.32 \approx 9.08 \text{ cm}^2$.
Step 3: Compare
Semicircle area $\approx 10.62 \text{ cm}^2$
Quarter-circle area $\approx 9.08 \text{ cm}^2$
✔ Yes, Marcus is correct. The semicircle’s area is greater.
❓ EXERCISES
11. Dan and Abi use different methods to work out the answer to this question.
a. Critique Dan and Abi’s methods.
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b. Which method do you think is better? Explain why.
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c. Use your favourite method to work out the radius of these circles. Give your answers correct to one decimal place (1 d.p.).
i. area = $35\ \text{cm}^2$
ii. area = $18.25\ \text{m}^2$
iii. area = $254\ \text{mm}^2$
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❓ EXERCISES
12. Here are six question cards and six answer cards.
a. Using estimation, match each question card with the correct answer card. Do not use a calculator.
b. Use a calculator to check you have matched the cards correctly.
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Step-by-step working:
- A. Work out $C$ when $d = 15.21$. $C = \pi d = \pi \times 15.21 \approx 47.78$. ✅ Matches answer card v ($47.78$).
- B. Work out $A$ when $d = 11.85$. $r = \tfrac{11.85}{2} = 5.925$, $A = \pi r^2 = \pi \times (5.925)^2 \approx 110.29$. ✅ Matches answer card i ($110.29$).
- C. Work out $C$ when $r = 9.5$. $C = 2 \pi r = 2 \pi \times 9.5 = 59.69 \approx 59.7$. ✅ Matches answer card vi ($59.7$).
- D. Work out $A$ when $r = 7.28$. $A = \pi r^2 = \pi \times (7.28)^2 \approx 166.50$. ✅ Matches answer card iii ($166.50$).
- E. Work out $d$ when $C = 87.2$. $C = \pi d \;\;\Rightarrow\;\; d = \tfrac{C}{\pi} = \tfrac{87.2}{\pi} \approx 27.8$. ✅ Matches answer card iv ($27.8$).
- F. Work out $r$ when $A = 304.09$. $A = \pi r^2 \;\;\Rightarrow\;\; r = \sqrt{\tfrac{A}{\pi}} = \sqrt{\tfrac{304.09}{\pi}} \approx 9.84$. ✅ Matches answer card ii ($9.84$).
Final matching:
- A → v ($47.78$)
- B → i ($110.29$)
- C → vi ($59.7$)
- D → iii ($166.50$)
- E → iv ($27.8$)
- F → ii ($9.84$)
13. The area of a circular pond is $21.5\,\text{m}^2$.
Work out the circumference of the pond.
Give your answer correct to the nearest centimetre.
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Use $A=\pi r^2$ to find the radius, then $C=2\pi r$.
1)$r = \sqrt{\dfrac{A}{\pi}} = \sqrt{\dfrac{21.5}{\pi}} \approx 2.616\,\text{m}$
2)$C = 2\pi r \approx 2\pi\times 2.616 \approx 16.437\,\text{m}$
Nearest centimetre: $16.437\,\text{m} = 1643.7\,\text{cm} \approx \mathbf{16.44\,\text{m}}$ (i.e., $\mathbf{1644\,\text{cm}}$).
❓ EXERCISES
14. The circumference of a circular lawn is $32.56\ \text{m}$. Work out the area of the lawn. Give your answer correct to the nearest square metre.
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15. This is part of Dirk’s classwork.
a. Critique this method of writing the answers. Does it have any advantages or disadvantages?
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b. Answer these questions. Write your answers in terms of $\pi$.
i. Work out the circumference of a circle with diameter $25\ \text{mm}$.
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ii. Work out the area of a circle with radius $12\ \text{mm}$.
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iii. Work out the circumference of a circle with radius $22.5\ \text{cm}$.
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iv. Work out the area of a circle with diameter $40\ \text{cm}$.
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c. The diagram shows a semicircle. Show that:
i. The area of the semicircle is $72\pi\ \text{m}^2$.
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ii. The perimeter of the semicircle is $12\pi + 24\ \text{m}$.