Gaokao Exam Solid Geometry Proofs and Calculations 2

Question 1

A regular triangular prism has base edge length 4 and height 6. What is its total surface area?

Accepted Answer

72 + 8√3

Answer

Two equilateral triangle bases: 2 × (√3/4 × 4²) = 8√3. Three rectangular lateral faces: 3 × (4 × 6) = 72. Total = 72 + 8√3.

Question 2

In a rectangular box (cuboid) with dimensions 3 × 4 × 12, what is the length of the space diagonal?

Accepted Answer

13

Answer

Space diagonal d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.

Question 3

A sphere is inscribed in a regular tetrahedron with edge length a. Which formula gives the radius of the inscribed sphere?

Accepted Answer

r = a√6/12

Answer

For a regular tetrahedron with edge a, the inradius r = a/(2√6) = a√6/12.

Question 4

The dihedral angle between two adjacent faces of a regular tetrahedron is approximately:

Accepted Answer

arccos(1/3) ≈ 70.5°

Answer

The dihedral angle of a regular tetrahedron is arccos(1/3) ≈ 70.53°. This can be derived by finding the angle between two face normal vectors.

Question 5

A cone has base radius r and slant height l. What is the lateral surface area?

Accepted Answer

πrl

Answer

The lateral surface area of a cone is πrl, where r is the base radius and l is the slant height (斜高). This comes from unrolling the lateral surface into a sector of a circle.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

Gaokao Exam Solid Geometry Proofs and Calculations 2