Gaokao Exam Solid Geometry Proofs and Calculations 2 — Questions and Answers

Question 1: A regular triangular prism has base edge length 4 and height 6. What is its total surface area?

72 + 8√3 (Correct answer)
48 + 8√3
72 + 4√3
48 + 4√3

Correct answer: 72 + 8√3

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

A regular triangular prism has two equilateral triangular bases and three rectangular lateral faces. Base area (equilateral triangle, side a=4): A_base = (√3/4)a² = (√3/4)(16) = 4√3. Two bases: 2 × 4√3 = 8√3. Lateral area: 3 rectangles each of dimensions 4 × 6: 3 × 24 = 72. Total surface area = 72 + 8√3 square units.

Question 2: In a rectangular box (cuboid) with dimensions 3 × 4 × 12, what is the length of the space diagonal?

13 (Correct answer)
√193
√169 = 13
5

Correct answer: 13

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

The space diagonal of a rectangular box (cuboid) with length l, width w, and height h is given by d = √(l² + w² + h²). With dimensions 3, 4, and 12: d = √(9 + 16 + 144) = √169 = 13. This is a clean answer because 3, 4, 12, 13 form a Pythagorean-related set (first: 3²+4²=25=5², then 5²+12²=169=13²). Recognizing these Pythagorean triples speeds up calculations in Gaokao solid geometry.

Question 3: A sphere is inscribed in a regular tetrahedron with edge length a. Which formula gives the radius of the inscribed sphere?

r = a√6/12 (Correct answer)
r = a/2
r = a√3/6
r = a/4

Correct answer: r = a√6/12

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

A regular tetrahedron has all four equilateral triangular faces equal. The inradius (radius of inscribed sphere) can be derived using the volume formula: V = (√2/12)a³ and surface area S = √3 a². Since V = (1/3)r × S: r = 3V/S = 3(√2/12)a³/(√3 a²) = (√2/4)a/√3 = a√2/(4√3) = a√6/12. This is a standard formula for Chinese high school solid geometry (立体几何).

Question 4: The dihedral angle between two adjacent faces of a regular tetrahedron is approximately:

arccos(1/3) ≈ 70.5° (Correct answer)
60°
90°
120°

Correct answer: arccos(1/3) ≈ 70.5°

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.

For a regular tetrahedron with vertices at convenient coordinates, the dihedral angle θ between any two adjacent faces satisfies cos θ = 1/3, giving θ = arccos(1/3) ≈ 70.53°. This can be derived by: (1) placing the tetrahedron with one face in the xy-plane, (2) finding the outward normal vectors of two adjacent faces, (3) computing the angle between them (and subtracting from 180° for the interior dihedral angle). This result is distinct from the 60° equilateral triangle angles within each face.

Question 5: A cone has base radius r and slant height l. What is the lateral surface area?

πrl (Correct answer)
πr²
2πrl
πr²l

Correct answer: πrl

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.

The lateral surface area of a cone is derived by 'unrolling' it into a flat sector. The sector has: radius = l (slant height), arc length = 2πr (circumference of base circle). Area of sector = (arc/full circumference) × πl² = (2πr/2πl) × πl² = πrl. Total surface area = lateral area + base area = πrl + πr². The slant height l = √(r² + h²), where h is the vertical height. These formulas are essential for Gaokao solid geometry calculations.

Question 6: In the Gaokao solid geometry proof format, what is the standard first step when asked to prove that a line is perpendicular to a plane?

Show that the line is perpendicular to two distinct lines within the plane that intersect at a point (Correct answer)
Show that the line is parallel to another perpendicular line
Measure the angle directly
Show that the line's projection onto the plane is a point

Correct answer: Show that the line is perpendicular to two distinct lines within the plane that intersect at a point

The standard theorem (三垂线定理's basis): a line ⊥ plane iff it is perpendicular to every line in the plane. The practical test: if a line is perpendicular to two intersecting lines in the plane, then it is perpendicular to the plane.

The standard criterion for proving a line ⊥ plane in Chinese high school solid geometry (线面垂直判定定理): 'If a line is perpendicular to two intersecting lines that lie in a plane, then the line is perpendicular to that plane.' The proof steps are: (1) identify two intersecting lines in the plane, (2) prove the target line is perpendicular to each, (3) invoke the theorem to conclude line ⊥ plane. This theorem is the foundation of most line-plane perpendicularity proofs in Gaokao solid geometry and must be stated explicitly when used.

Gaokao Exam Solid Geometry Proofs and Calculations 2

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