Gaokao Exam Solid Geometry Proofs and Calculations 1 — Questions and Answers
Question 1: A rectangular box (cuboid) has dimensions 2 × 4 × 4. What is the radius of its circumscribed sphere?
- 3 (Correct answer)
- 2√6
- 2√3
- √10
Correct answer: 3
The space diagonal of a cuboid with dimensions a × b × c is √(a²+b²+c²) = √(4+16+16) = √36 = 6. The circumscribed sphere's diameter equals the space diagonal, so R = 6/2 = 3.
Question 2: A regular tetrahedron has edge length 2. What is its volume?
- 4√3/3
- 2√2/3 (Correct answer)
- √2/3
- 8√3/3
Correct answer: 2√2/3
The volume of a regular tetrahedron with edge length a is V = a³√2/12. Substituting a = 2: V = 8√2/12 = 2√2/3.
Question 3: A regular quadrilateral (square) pyramid has a base side length of 4 and a height of 2√2. What is the angle between a lateral edge and the base?
- 30°
- 60°
- 45° (Correct answer)
- arctan(√3)
Correct answer: 45°
The half-diagonal of the square base is (4√2)/2 = 2√2. The lateral edge length = √(height² + half-diagonal²) = √(8 + 8) = 4. Thus cos θ = (2√2)/4 = √2/2, giving θ = 45°.
Question 4: A sphere of radius r is inscribed in a right circular cylinder. What is the ratio of the sphere's total surface area to the cylinder's total surface area?
- 1/2
- 3/4
- 1
- 2/3 (Correct answer)
Correct answer: 2/3
The inscribed sphere has radius r, so the cylinder has radius r and height 2r. Sphere SA = 4πr². Cylinder total SA = 2πr² + 2πr(2r) = 6πr². The ratio is 4πr²/6πr² = 2/3.
Question 5: A right circular cone has a base circumference of 6Ï€ and a slant height of 5. What is its lateral surface area?
- 9Ï€
- 12Ï€
- 15Ï€ (Correct answer)
- 18Ï€
Correct answer: 15Ï€
From 2πr = 6π we get base radius r = 3. The lateral surface area formula is πrl = π(3)(5) = 15π.
Question 6: A frustum (truncated cone) has a top base radius of 2, a bottom base radius of 4, and a height of 3. What is its volume?
- 24Ï€
- 28Ï€ (Correct answer)
- 32Ï€
- 36Ï€
Correct answer: 28Ï€
Using the frustum volume formula V = (πh/3)(R² + Rr + r²) = (π·3/3)(16 + 8 + 4) = π(28) = 28π.
Gaokao Exam Solid Geometry Proofs and Calculations 1
A rectangular box (cuboid) has dimensions 2 × 4 × 4.
What is the radius of its circumscribed sphere?