The figure shows two circles passing through each other’s centres. Find the area of the region enclosed by the two circles in terms of the common radius r.
Answer: The shaded common region = 2 segments, each with central angle 120°: 2(πr23 − √34r2) = (2π3 − √32)r2 ≈ 1.23r2. (The whole region covered by the two circles is 2πr2 minus this, (4π3 + √32)r2 ≈ 5.05r2.)
Step-by-step solution
Idea: As in the two-circle perimeter problem, triangles ABC and ABD are equilateral, so the chord CD subtends 120° at each centre. The shaded lens is two equal segments.
- AB = AC = BC = r, so △ABC is equilateral; so is △ABD. Hence ∠CAD = 60° + 60° = 120°, and similarly ∠CBD = 120°.1 mark
- Sector ACD of circle A (120°) = 120360πr2 = πr23. △ACD has AC = AD = r and height from A to CD equal to r2 (half of AB), with CD = √3r; its area is 12 × √3r × r2 = √34r2.1 mark
- Segment of circle A beyond CD = πr23 − √34r2. The shaded region is this segment plus the equal one from circle B.1 mark
- Shaded area = 2(πr23 − √34r2) = (2π3 − √32)r2 ≈ (2.094 − 0.866)r2 ≈ 1.23r2.1 mark
The shaded region common to both circles has area (2π/3 − √3/2)r² ≈ 1.23r². (The total area covered by the two circles together is (4π/3 + √3/2)r² ≈ 5.05r².)
Check: 1.23r2 is less than half a circle (1.57r2), which fits the picture ✓.
Answer to write in the exam
△ABC, △ABD equilateral ⇒ ∠CAD = ∠CBD = 120°
Sector ACD = 120°360°πr2 = πr23
ar(△ACD) = 12 × √3r × r2 = √34r2
Segment = πr23 − √34r2
Shaded = 2 × segment
∴ Area = (2π3 − √32)r2 ≈ 1.23r2
Common mistakes that cost marks
- Using 60° instead of 120° for the sector.
- Adding the triangle instead of subtracting it from the sector.
- Using 12r2 for the triangle as if it were right-angled.
How this can come in the exam
MCQ (1 mark)
Two circles of radius 6 cm pass through each other’s centres. The length of their common chord is
- 6 cm
- 6√3 cm
- 3√3 cm
- 12 cm
Show answer
(B) 6√3 cm
CD = √3r = 6√3 cm.
Try one yourself
Find the area of the common region of two circles of radius 10 cm that pass through each other’s centres. (Use π = 3.14, √3 = 1.73.)
Show answer
(2 × 3.143 − 1.732) × 100 = (2.0933 − 0.865) × 100 ≈ 122.83 cm2.
More questions like this
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