Solid Geometry Proofs and Calculations Flashcards

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

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.

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

Answer: 13

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

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

Answer: r = a√6/12

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

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

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.

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

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.

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?

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.