Every circle has the same ratio of circumference to diameter. That ratio is π, about 3.14159.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The two formulas
Circumference is 2πr and area is πr². The area formula squares the radius; the circumference formula does not.
Telling them apart
Area is measured in square units, so it must involve a squared length. That check settles which formula you want.
Arc length
An arc is a fraction of the circumference. Take the central angle over 360° and multiply.
Sector area
A sector is the matching fraction of the area, using the same angle over 360°.
Radius, not diameter
Both formulas use the radius. Substituting a diameter doubles or quadruples the answer.
Where π comes from
Divide any circle's circumference by its diameter and the answer is always about 3.14159. That constant ratio was discovered by measurement and is the same for every circle.
Circumference and area answer different questions
Circumference is a length in centimetres; area is a surface in square centimetres. The formulas look similar and are not interchangeable, so read which the question wants.
Arcs and sectors are fractions of the whole
The central angle over 360° gives the fraction. Multiply by the circumference for arc length or by the area for sector area — one idea covering both formulas.
Check radius against diameter
The area formula uses the radius. Substituting the diameter gives an answer four times too large, because the halving is then squared. Halve it before doing anything else.
Step 2: Try It Yourself
Tap and try it out.
- Diameter8 cm
- Circumference25.13 cm
- Circumference ÷ diameter3.14
Change the size. The circle gets bigger, but circumference divided by diameter stays at about 3.14 every time. That number is π.
Step 3: Watch an Example
One step at a time.
Watch Sana Find an Arc
A circle of radius 10 has a sector with a central angle of 90°.
- Step 1
The full circumference is 2π(10) = 20π.
Step 4: Your Turn
Practice makes it stick.
The Circumference
Problem 1 of 2
A circle has radius 10. What is the circumference divided by π?
The Area
Problem 2 of 2
Same circle. What is the area divided by π?
Measure the Circle
1 of 8
Radius 5. Circumference divided by π?
2 of 8
Radius 5. Area divided by π?
3 of 8
Diameter 12. What is the radius?
4 of 8
A 90° sector is what fraction of the circle, as a decimal?
5 of 8
A 120° sector is what fraction, to three decimal places?
6 of 8
Radius 6, 180° sector. Arc length divided by π?
7 of 8
Sort each formula by what it measures.
Tap something to move it.
- Empty
- Empty
8 of 8
Radius 3. Area divided by π?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Radius 7. What is the area divided by π?
Question 2 of 2
How can you tell which formula gives area?
What You Learned
- Circumference is 2πr and area is πr².
- An arc or sector is the fraction angle ÷ 360 of the whole.
- Both formulas use the radius, never the diameter.