Arc Length and Sector Area
This topic covers calculations for sectors, which are wedge-shaped portions of a circle. You will need to calculate a sector's area, the length of its curved edge (arc length), and its total perimeter by treating it as a fraction of a full circle.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A sector is a fraction of a circle, determined by its central angle. The fraction is always the angle, θ, divided by the 360 degrees in a full circle.
- The area of a sector is this fraction applied to the full circle's area: (θ/360) × πr².
- The arc length is the same fraction applied to the full circle's circumference: (θ/360) × 2πr.
- The perimeter of a sector is the sum of the arc length and the two radii that form its straight sides: Perimeter = Arc Length + 2r.
- Problems can require you to work backwards, for example, using a given area or arc length to find the radius or the angle.
Diagram
Formulae
Area = (theta / 360) × pi × r2 To calculate the area of a sector when you know the angle (theta) and radius (r).
Arc Length = (theta / 360) × 2 × pi × r To calculate the length of the curved edge of a sector when you know the angle (theta) and radius (r).
Perimeter = 2*r + (theta / 360) × 2 × pi × r To calculate the total length of the boundary of a sector, including its two straight sides.
Definitions
- Sector
- A region of a circle enclosed by two radii and the arc connecting them, much like a slice of pizza.
- Arc Length
- The distance along the curved edge of a sector.
- Sector Angle (θ)
- The angle formed at the center of the circle by the two radii of the sector, typically measured in degrees.
Worked example
A sector is cut from a circle of radius 6 cm. Its area is 10π cm². What is the perimeter of the sector?
- 1
Start with the area formula:
Area = (theta / 360) × pi × r² - 2
Substitute the known values:
10π = (theta / 360) × pi × 6² - 3
Simplify the equation:
10π = (theta / 360) × 36π - 4
Cancel π from both sides:
10 = (theta / 360) × 36 - 5
Simplify the fraction:
10 = theta × (36/360) = theta / 10 - 6
Solve for the angle:
theta = 10 × 10 = 100 degrees - 7
Now, calculate the arc length using the angle:
Arc Length = (100 / 360) × 2 × pi × 6 - 8
Simplify the calculation:
Arc Length = (10/36) × 12π = (5/18) × 12π = (5/3) × 2π = 10π/3 cm - 9
Finally, find the perimeter:
Perimeter = 2r + Arc Length - 10
Substitute the values:
Perimeter = 2(6) + 10π/3 = 12 + 10π/3 cm
Answer: 12 + 10π/3 cm
Common mistakes
- ×Mixing up the area and circumference formulas. Forgetting that area involves r² while arc length (part of circumference) involves 2πr.
- ×Calculating only the arc length when asked for the perimeter. Remember the perimeter also includes the two straight radii.
- ×Making arithmetic errors when simplifying the fraction (θ/360) or when squaring the radius. Cancel common factors early to keep the numbers manageable.
No-calculator tips
- ✓Always simplify the fraction θ/360 before multiplying. For example, 120/360 simplifies to 1/3, which is much easier to work with.
- ✓Keep π as a symbol throughout your calculation. This avoids dealing with decimals and allows you to cancel π if it appears on both sides of an equation.
- ✓If you have to square a number, do it carefully. For example, 15² = 225. Break down larger calculations, e.g., 324 / 9 = (270 + 54) / 9 = 30 + 6 = 36.