Single correct option
A sphere is inscribed in a right circular cone of base radius 6 cm and height 8 cm, touching the base and the curved surface. What is the ratio of the volume of the sphere to the volume of the cone?
- A. 3 : 8
- B. 1 : 2
- C. 9 : 32
- D. 3 : 4
Show answer and solution
Answer: (A) 3 : 8
Take the vertical cross-section through the axis: an isosceles triangle with base 12, height 8 and equal sides √(6² + 8²) = 10. The sphere appears as the incircle of this triangle.
Inradius r = Area/semi-perimeter = (1/2 × 12 × 8)/((10 + 10 + 12)/2) = 48/16 = 3 cm.
Volume of sphere = (4/3)π(27) = 36π. Volume of cone = (1/3)π(36)(8) = 96π.
Ratio = 36π : 96π = 3 : 8.
Tip: Any 'sphere inside cone' problem becomes an incircle problem in the axial cross-section.
Inradius r = Area/semi-perimeter = (1/2 × 12 × 8)/((10 + 10 + 12)/2) = 48/16 = 3 cm.
Volume of sphere = (4/3)π(27) = 36π. Volume of cone = (1/3)π(36)(8) = 96π.
Ratio = 36π : 96π = 3 : 8.
Tip: Any 'sphere inside cone' problem becomes an incircle problem in the axial cross-section.
Saved on this device. Sign in or create a free account to keep it on every device.