Radians Arc Length and Sectors
This topic introduces radians as an alternative to degrees for measuring angles, which is the standard in higher-level mathematics. For the ESAT, you must be able to use radians to calculate the arc length, sector area, and segment area of a circle without a calculator.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The fundamental relationship is π radians = 180°. All angle conversions stem from this fact.
- The formulae for arc length and sector area are only valid when the angle is measured in radians.
- The area of a segment is found by subtracting the area of a triangle from the area of a sector.
- Memorise the common angle conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2.
- The perimeter of a sector is the sum of two radii and the arc length (2r + l).
Diagram
Formulae
l = r × theta To calculate the arc length (l) of a sector with radius (r) and angle (theta) in RADIANS.
Asector = (1/2) × r2 × theta To calculate the area of a sector with radius (r) and angle (theta) in RADIANS.
Asegment = (1/2) × r2 × (theta - sin(theta)) To calculate the area of a segment. This combines the sector area and the area of the inner triangle (A = (1/2)ab*sin(C)). Angle theta must be in RADIANS.
Definitions
- Radian
- The angle at the centre of a circle where the arc length is equal to the radius. One full circle contains 2π radians.
- Sector
- A region of a circle enclosed by two radii and the arc connecting them, similar in shape to a slice of pizza.
- Segment
- A region of a circle enclosed by a chord and the arc connecting the chord's endpoints.
Worked example
A sector of a circle has radius 6 cm and a perimeter of (12 + 2π) cm. Calculate the exact area of the sector.
- 1
The perimeter of a sector is given by P = 2r + l, where l is the arc length.
- 2
Substitute the given values:
12 + 2π = 2*(6) + l - 3
Solve for the arc length:
12 + 2π = 12 + l, so l = 2π cm - 4
Use the arc length formula l = r × theta to find the angle in radians.
- 5
Substitute known values:
2π = 6 × theta, which gives theta = 2π / 6 = π/3 radians - 6
Now use the sector area formula A = (1/2) × r2 × theta.
- 7
Calculate the area:
A = (1/2) × 62 × (π/3) = (1/2) × 36 × (π/3) - 8
Simplify the expression:
A = 18 × (π/3) = 6π cm2
Answer: 6π cm2
Common mistakes
- ×Using degrees instead of radians in the standard formulae. This is a frequent error that leads to an answer being incorrect by a large factor, often involving π/180.
- ×Confusing the formula for a segment's area. A common mistake is to calculate (1/2) × r2 × (theta - theta) instead of (1/2) × r2 × (theta - sin(theta)), effectively getting zero.
- ×Making arithmetic errors with π and fractions. Keep π as a symbol until the very end and be methodical with fraction multiplication and simplification to avoid calculation slips.
No-calculator tips
- ✓To convert from degrees to radians, multiply by π/180. To convert from radians to degrees, multiply by 180/π. Think of it as 'cancelling' the units you don't want.
- ✓When simplifying expressions like (1/2) × r2 × theta, deal with the numerical parts first. For example, in (1/2) × 82 × (π/4), calculate (1/2) × 64 = 32, then 32/4 = 8. The answer is 8π.
- ✓If a question involves perimeters or areas combining straight lines and curves, draw a clear diagram and label each part (radii, arc length, chords) before you start calculating.