Circumference of a Circle
The circumference of a circle is the distance around its edge. It depends on the radius (r), which is the distance from the centre to any point on the circle.
The formula for circumference is: C = 2πr, where π (pi) is approximately 3.1416 but often left as π in answers for exactness.
To calculate circumference, substitute the radius into the formula and multiply by 2 and π.
- Formula: C = 2πr
- r = radius of the circle
- π is a constant approximately equal to 3.1416
Area of a Circle
The area of a circle is the space enclosed within its boundary. It depends on the radius squared.
The formula for area is: A = πr². This means you square the radius and multiply by π.
Expressing answers in terms of π keeps them exact, which is often preferred in exams.
- Formula: A = πr²
- r = radius of the circle
- Area measures the space inside the circle
Arc Length of a Sector
An arc is a part of the circumference of a circle. The length of an arc depends on the angle θ of the sector it subtends at the centre.
Since the full circumference corresponds to 360°, the arc length L for an angle θ is a fraction θ/360 of the full circumference.
The formula is: L = (θ/360) × C = (θ/360) × 2πr. This allows calculation of arc length when θ divides 360° exactly.
- Arc length is part of the circumference
- Formula: L = (θ/360) × 2πr
- θ is the sector angle in degrees
Sector Area
A sector is the portion of a circle enclosed by two radii and the arc between them.
The area of a sector is a fraction θ/360 of the total area of the circle, where θ is the sector angle in degrees.
The formula is: S = (θ/360) × A = (θ/360) × πr². This calculates the sector area exactly when θ divides 360°.
- Sector area is part of the circle's area
- Formula: S = (θ/360) × πr²
- θ is the sector angle in degrees
Worked Examples
Example 1: Calculate the circumference of a circle with radius 7 cm, expressing the answer in terms of π.
Using C = 2πr, substitute r = 7 cm: C = 2 × π × 7 = 14π cm.
Example 2: Find the area of the same circle, expressing the answer in terms of π: A = πr² = π × 7² = 49π cm², which is exact and preferred in many cases for precision rather than decimal approximation.
- Circumference = 14π cm
- Area = 49π cm²
Worked Examples for Arc Length and Sector Area
Example 3: Find the arc length of a sector with angle 90° in a circle of radius 7 cm.
First, find the circumference: C = 14π cm. Then, L = (90/360) × 14π = (1/4) × 14π = 3.5π cm.
Example 4: Find the sector area for the same sector: S = (90/360) × 49π = (1/4) × 49π = 12.25π cm².
- Arc length = 3.5π cm
- Sector area = 12.25π cm²
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Calculate circumference | Use C = 2πr, substitute radius | r=7 cm → C=14π cm | Write formula, substitute, keep π for exactness |
| Calculate area | Use A = πr², substitute radius | r=7 cm → A=49π cm² | Show squaring radius, keep π in answer |
| Calculate arc length | Find fraction θ/360, multiply by circumference | θ=90°, L=(90/360)×14π=3.5π cm | Express arc length as fraction of circumference |
| Calculate sector area | Find fraction θ/360, multiply by area | θ=90°, S=(90/360)×49π=12.25π cm² | Express sector area as fraction of total area |
Active recall
Quick check
1. What is the formula for the circumference of a circle with radius r?
C = 2πr, where r is the radius.
2. How do you calculate the area of a sector with angle θ in a circle of radius r?
Sector area S = (θ/360) × πr².
3. If a sector angle is 60°, what fraction of the circle's circumference does its arc length represent?
60/360 = 1/6 of the circumference.
4. Calculate the arc length of a sector with angle 120° in a circle of radius 5 cm, expressing the answer in terms of π.
C = 2π × 5 = 10π cm; arc length L = (120/360) × 10π = (1/3) × 10π = 10π/3 cm.