โญ Circumference of a Circle
Cambridge Lower Secondary ยท Grade 7 ยท Geometry & Measure
Using Diameter
C = ฯd e.g. d = 10 cm
C = ฯ ร 10 โ 31.42 cm
Using Radius
C = 2ฯr e.g. r = 7 m
C = 2 ร ฯ ร 7 โ 43.98 m
The Magic Ratio
C รท d = ฯ for every circle
ฯ โ 3.14159...
Watch circumference unroll! โญโก๏ธ๐
What you'll learn:
- Parts of a circle: radius, diameter, chord, arc, sector, tangent, circumference
- Pi (ฯ): the magical ratio C/d = ฯ โ 3.14159
- Formulas: C = ฯd and C = 2ฯr
- Finding circumference given radius or diameter
- Working backwards: find r or d given circumference
- Arc length: (ฮธ/360) ร 2ฯr
- Real contexts: wheel rotations, running tracks, gears
๐ Learn: Circumference of a Circle
Part 1: Parts of a Circle
Every circle has special parts. Click a part below to highlight it on the diagram.
| Part | Definition |
|---|
| Radius (r) | Distance from centre to edge. All radii of a circle are equal. |
| Diameter (d) | Distance across the circle through the centre. d = 2r. |
| Chord | A straight line joining two points on the circumference (not necessarily through the centre). |
| Arc | Part of the circumference between two points. |
| Sector | A "pizza slice" โ the region between two radii and an arc. |
| Tangent | A line that just touches the circumference at exactly one point, at 90ยฐ to the radius. |
| Circumference | The perimeter of the circle โ the total distance around it. |
Part 2: Pi (ฯ) โ The Magic Ratio
For any circle, if you divide the circumference by the diameter, you always get the same number: ฯ (pi).
C รท d = ฯ โ 3.14159...
๐ ฯ is an irrational number โ its decimal goes on forever with no repeating pattern.
ฯ โ 3.14159265358979...
For most problems, use your calculator's ฯ button, or approximate as 3.14 or 22/7.
Ancient civilisations measured this: a wheel with diameter 1 m rolls exactly ฯ metres in one full turn!
๐ฅง Fun memory trick: Think of a pie โ Pi-E ... 3.1415926... "May I have a large container of coffee?" (count letters: 3ยท1ยท4ยท1ยท5ยท9ยท2ยท6)
Part 3: The Circumference Formulas
Since C/d = ฯ, we can rearrange to get the two key formulas:
C = ฯd
or
C = 2ฯr
Both are the same! Since d = 2r, replacing d with 2r gives C = ฯ(2r) = 2ฯr.
Use C = ฯd when you're given the diameter.
Use C = 2ฯr when you're given the radius.
๐ก Remember: d = 2r and r = d/2. You can always switch between them!
Part 4: Working Backwards
If you know the circumference, you can find the radius or diameter by rearranging.
๐ From C = ฯd โ d = C รท ฯ
๐ From C = 2ฯr โ r = C รท (2ฯ)
Example: C = 50 cm. Find d. d = 50 รท ฯ โ 50 รท 3.14159 โ 15.92 cm
Example: C = 88 m. Find r. r = 88 รท (2ฯ) โ 88 รท 6.2832 โ 14.01 m
๐ Rearranging: whatever you divide by in the formula, you divide C by that to go backwards!
Part 5: Arc Length
An arc is a fraction of the full circumference. The fraction depends on the angle at the centre (ฮธ).
Arc length = (ฮธ/360) ร 2ฯr
๐ A full circle has 360ยฐ. A 90ยฐ arc is 90/360 = 1/4 of the circumference.
Example: r = 5 cm, ฮธ = 120ยฐ. Arc = (120/360) ร 2 ร ฯ ร 5 = (1/3) ร 31.42 โ 10.47 cm
Example: r = 10 m, ฮธ = 45ยฐ. Arc = (45/360) ร 2 ร ฯ ร 10 = (1/8) ร 62.83 โ 7.85 m
๐ Think pizza slices: a 180ยฐ arc is half the circumference, a 90ยฐ arc is a quarter, and so on!
๐ก Worked Examples
Example 1: Find Circumference Given Diameter
A circular plate has a diameter of 24 cm. Find its circumference. Give your answer to 2 decimal places.
Step 1: Write the formula. C = ฯd
Step 2: Substitute. C = ฯ ร 24
Step 3: Calculate. C = 75.398... cm
Answer: C โ 75.40 cm
๐ Always write the formula first, then substitute values, then calculate.
Example 2: Find Circumference Given Radius
A circular running track has radius 35 m. Find its circumference. Give your answer in terms of ฯ.
Step 1: Write the formula. C = 2ฯr
Step 2: Substitute. C = 2 ร ฯ ร 35
Step 3: Simplify (leave in terms of ฯ). C = 70ฯ m
Answer: C = 70ฯ m โ 219.91 m
๐ก "In terms of ฯ" means keep ฯ as a symbol โ don't press the ฯ button! Just write 70ฯ.
Example 3: Find Radius Given Circumference
A bicycle wheel has a circumference of 188.5 cm. Find the radius. (Use ฯ โ 3.14)
Step 1: Write the formula. C = 2ฯr
Step 2: Substitute. 188.5 = 2 ร 3.14 ร r
Step 3: Simplify. 188.5 = 6.28 ร r
Step 4: Divide both sides by 6.28. r = 188.5 รท 6.28
Answer: r = 30 cm
๐ When working backwards, isolate r by dividing by 2ฯ.
Example 4: Arc Length Problem
A sector has radius 9 cm and angle 80ยฐ at the centre. Find the arc length. Give your answer to 1 decimal place.
Step 1: Write the formula. Arc = (ฮธ/360) ร 2ฯr
Step 2: Substitute. Arc = (80/360) ร 2 ร ฯ ร 9
Step 3: Calculate fraction. 80/360 = 2/9
Step 4: Arc = (2/9) ร 56.549... = 12.566... cm
Answer: Arc โ 12.6 cm
๐ The angle fraction (80/360) tells you what share of the full circle this arc represents.
โญ Circle Visualizer
Rolling Circle โ Circumference Unrolled
Watch the circle roll forward and its circumference unroll into a straight line equal to ฯd.
Normal
Live Circle Builder
Drag the slider to change the radius. Watch r, d, and C update instantly!
5 cm
โ๏ธ Exercise 1: Name Circle Parts
Match each description to the correct circle part. Type the name in each box.
Words: radius, diameter, chord, arc, sector, tangent, circumference
โ๏ธ Exercise 2: Find Circumference
Calculate the circumference. Round to 2 decimal places (use ฯ โ 3.14159).
โ๏ธ Exercise 3: Find Radius or Diameter
Given the circumference, find the radius or diameter. Round to 2 decimal places.
๐ Exercise 4: Arc Length
Find the arc length for each sector. Round to 2 decimal places.
๐ Exercise 5: Real-World Problems
Apply circumference to real-life contexts. Round to 2 decimal places unless told otherwise.
๐ Challenge Questions
These questions require multi-step reasoning. Give answers to 2 d.p. unless stated.