A chord of a circle of radius r subtends an angle of 60° at the centre of the circle. Show that the area of the corresponding minor segment of the circle is equal to πr2(16 − √34).
Step-by-step solution
Idea: Segment = sector − triangle. With a 60° angle, triangle OAB has two radii as sides and a 60° angle between them, so it is equilateral with side r.
- Sector OAB = πr2 × 60°360° = πr26.1 mark
- OA = OB = r and ∠AOB = 60°, so ∠OAB = ∠OBA = 60° and △OAB is equilateral with side r. Its area = √34r2.1 mark
- Minor segment = πr26 − √34r2 = r2(π6 − √34) ≈ 0.0906r2. This is what the question intends; as printed, πr2(16 − √34) would equal πr26 − π√34r2, a negative number, so the π must multiply only the 16.1 mark
Check: With r = 15, π = 3.14, √3 = 1.73: 225 × (0.5233 − 0.4325) ≈ 20.44 cm2, the same as the 15 cm, 60° segment worked out with numbers ✓.
Answer to write in the exam
Sector = πr2 × 60°360° = πr26
OA = OB = r, ∠AOB = 60° ⇒ △OAB equilateral
ar(△OAB) = √34r2
Minor segment = πr26 − √34r2
∴ Minor segment = r2(π6 − √34)
Common mistakes that cost marks
- Taking the triangle as right-angled and using 12r2.
- Factorising πr2 out of both terms, which wrongly puts π on the triangle’s area.
- Adding the triangle to the sector instead of subtracting.
How this can come in the exam
A chord of a circle of radius r subtends 90° at the centre. The area of the minor segment is
- r2(π4 − 12)
- πr2(14 − 12)
- πr24
- r2(π2 − 1)
Show answer
(A) r2(π4 − 12)
Sector πr24 minus right triangle 12r2.
Try one yourself
Find the area of the minor segment of a circle of radius 21 cm cut off by a chord subtending 90° at the centre. (Use π = 227.)
Show answer
14 × 1386 − 12 × 441 = 346.5 − 220.5 = 126 cm2.
More questions like this
- An equilateral triangle is inscribed in a circle of radius r. Show that the ratio of the area of the triangle to the area of the circle is equal to 3√34π ≈ 0.413.
- A square is inscribed in a circle of radius r. Show that the ratio of the area of the square to the area of the circle is equal to 2π ≈ 0.637.
- A hexagon is inscribed in a circle of radius r. Show that the ratio of the area of the hexagon to the area of the circle is equal to 3√32π ≈ 0.827. Can you see why the answer is exactly twice the answer for the inscribed equilateral triangle?
- Identities in algebra can sometimes be shown as area relationships. For example: The figure shown corresponds to the identity (a + b)2 = a2 + 2ab + b2. Do you see how? Draw figures corresponding to the identities (a + b)(a − b) = a2 − b2 and (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
- An isosceles triangle has perimeter 40 cm; the equal sides are 15 cm each. Find the area of the triangle.