Circumference and Area of a Circle
The circumference of a circle is the distance around its edge. It depends on the radius (r) and the constant π (pi), approximately 3.1416. The formula for circumference is:
\[ C = 2\pi r \] where C is the circumference and r is the radius.
The area of a circle is the space enclosed within its boundary. It is calculated using the radius and π as: \[ A = \pi r^2 \] where A is the area. Expressing answers in terms of π keeps them exact and avoids rounding errors.
- Circumference formula: \( C = 2\pi r \)
- Area formula: \( A = \pi r^2 \)
- Use radius (r) as the key measurement
- Express answers in terms of π for exactness
Calculating Arc Lengths
An arc is a portion of the circumference defined by a central angle \( \theta \) (in degrees). The arc length is the fraction of the total circumference corresponding to this angle. Since a full circle is 360°, the arc length \( L \) is:
\[ L = \frac{\theta}{360} \times 2\pi r \] This formula calculates the length of the minor arc when \( \theta \) is less than 180°. For example, if the central angle is 90°, the arc length is one quarter of the circumference.
For major arcs (angles greater than 180°), use the reflex angle \( 360° - \theta \) to find the arc length of the larger arc.
- Arc length formula: \( L = \frac{\theta}{360} \times 2\pi r \)
- \( \theta \) is the central angle in degrees
- Use \( 360° - \theta \) for major arcs
- Express arc length in terms of π when possible
Calculating Sector Areas
A sector is the region enclosed by two radii and the arc between them. Its area is a fraction of the total circle's area, proportional to the central angle \( \theta \). The sector area \( S \) is given by:
\[ S = \frac{\theta}{360} \times \pi r^2 \] This formula applies to minor sectors where \( \theta \) is less than 180°. For major sectors, use the reflex angle \( 360° - \theta \) to find the larger sector's area.
Expressing the sector area in terms of π keeps the answer exact.
- Sector area formula: \( S = \frac{\theta}{360} \times \pi r^2 \)
- \( \theta \) is the central angle in degrees
- Use \( 360° - \theta \) for major sectors
- Express sector area in terms of π
Distinguishing Minor and Major Sectors and Arcs
Minor sectors and arcs correspond to central angles less than or equal to 180°. Major sectors and arcs correspond to central angles greater than 180°. When given a minor angle \( \theta \), the major angle is \( 360° - \theta \).
Calculations must use the correct angle depending on whether the problem asks for the minor or major sector/arc. This distinction affects both arc length and sector area calculations.
Always check the context or diagram to identify which sector or arc is required.
- Minor sector/arc: central angle \( \leq 180° \)
- Major sector/arc: central angle \( > 180° \)
- Major angle = \( 360° - \theta \) if \( \theta \) is minor angle
- Use correct angle for accurate calculations
Worked Examples
Example 1: Calculate the circumference and area of a circle with radius 7 cm, expressing answers in terms of π.
Circumference: \[ C = 2\pi r = 2\pi \times 7 = 14\pi \text{ cm} \]
Area: \[ A = \pi r^2 = \pi \times 7^2 = 49\pi \text{ cm}^2 \]
- Circumference = \( 14\pi \) cm
- Area = \( 49\pi \) cm²
Worked Examples (continued)
Example 2: Find the length of an arc with central angle 60° in a circle of radius 10 cm.
Arc length: \[ L = \frac{60}{360} \times 2\pi \times 10 = \frac{1}{6} \times 20\pi = \frac{20\pi}{6} = \frac{10\pi}{3} \text{ cm} \]
Example 3: Calculate the area of the sector with central angle 60° in the same circle (radius 10 cm). Express in terms of π: \[ S = \frac{60}{360} \times \pi \times 10^2 = \frac{1}{6} \times 100\pi = \frac{100\pi}{6} = \frac{50\pi}{3} \text{ cm}^2 \]
- Arc length = \( \frac{10\pi}{3} \) cm
- Sector area = \( \frac{50\pi}{3} \) cm²
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Calculate circumference | Use formula \( C=2\pi r \) with radius | Radius 7 cm: \( 14\pi \) cm | Show substitution and express in terms of π |
| Calculate area | Use formula \( A=\pi r^2 \) | Radius 7 cm: \( 49\pi \) cm² | Write exact answer with π |
| Find arc length | Use \( L=\frac{\theta}{360} \times 2\pi r \) | \( \theta=60° \), radius 10 cm: \( \frac{10\pi}{3} \) cm | Identify minor/major arc angle |
| Find sector area | Use \( S=\frac{\theta}{360} \times \pi r^2 \) | \( \theta=60° \), radius 10 cm: \( \frac{50\pi}{3} \) cm² | Express answer in terms of π |
| Handle major sectors/arcs | Use angle \( 360° - \theta \) | If minor angle 60°, major angle 300° | Check which sector/arc is asked |
Active recall
Quick check
1. What is the formula for the circumference of a circle with radius r?
The circumference is \( C = 2\pi r \).
2. How do you calculate the arc length of a sector with central angle \( \theta \)?
Arc length \( L = \frac{\theta}{360} \times 2\pi r \).
3. How do you find the area of a sector with central angle \( \theta \)?
Sector area \( S = \frac{\theta}{360} \times \pi r^2 \).
4. If the minor sector angle is 70°, what angle do you use to find the major sector area?
Use \( 360° - 70° = 290° \) for the major sector.
5. Why express answers in terms of π?
Expressing answers in terms of π keeps them exact and avoids rounding errors.