Gaokao Exam Solid Geometry Proofs and Calculations 3

Question 1

A regular pyramid has a square base with side 6 and four equilateral triangular lateral faces. What is the height of the pyramid?

Accepted Answer

3√2

Answer

6

Answer

3

Answer

√18

Question 2

In a cube with edge length 2, what is the distance from the center of the cube to a vertex?

Accepted Answer

√3

Answer

2

Answer

√2

Answer

2√3

Question 3

When setting up a coordinate system (建系) for a solid geometry problem in the Gaokao, the best choice for origin and axes is typically:

Accepted Answer

Place the origin at a vertex where three mutually perpendicular edges meet, aligning axes along those edges

Answer

Place the origin at the centroid of the solid

Answer

Align axes with the longest diagonal

Answer

Use polar coordinates

Question 4

In solid geometry, the 'three perpendicular lines theorem' (三垂线定理) states: If a line (OA) in a plane (α) is the projection of an oblique line (PA) (where P is outside α), and OA ⊥ l (where l is any line in α through O), then:

Accepted Answer

PA ⊥ l

Answer

PA is parallel to l

Answer

PA ⊥ the plane α

Answer

OA ⊥ PA

Question 5

A sphere of radius R is circumscribed about a cube (the cube is inscribed in the sphere). What is the edge length of the cube in terms of R?

Accepted Answer

2R/√3 = 2R√3/3

Answer

R

Answer

R√2

Answer

R/2

Question 6

To find the dihedral angle along a common edge in a solid geometry problem, the most systematic method in Chinese high school is:

Accepted Answer

Find the half-plane angle by constructing perpendiculars from a point on the edge to each face, forming the dihedral angle's measurement angle

Answer

Measure it directly from a 3D figure

Answer

Use only the dot product formula without geometric construction

Answer

Assume it equals a face angle