Solid Geometry Advanced Solutions
16 advanced solid geometry problems on spheres, proofs, angles, distances, and comprehensive 3D reasoning.
16 problems
Problem 1: Prove that line is parallel to plane
In the right triangular prism , the lateral edges are perpendicular to the base plane. Assume and . Point is the midpoint of , and point is the midpoint of segm
Problem 2: Prove that plane
In , . Points and are the midpoints of and respectively, and . Fold along line to a new position in space, so that is the image of and . Prove that plane .
Problem 3: Prove that plane is perpendicular to plane
In tetrahedron , suppose plane and . Prove that plane is perpendicular to plane .
Problem 4: Prove that the two lateral faces plane and plane are perpendicular
In a right triangular prism , the lateral edges are perpendicular to the base plane, and . Prove that the two lateral faces plane and plane are perpendicular.
Problem 5: Find , where is the angle between planes and
In the right triangular prism , the lateral edges are perpendicular to the base plane. Assume and . Let be the midpoint of . Find , where is the angle between p
Problem 6: find , where is the angle between the line and the base plane
In a cube with edge length , find , where is the angle between the line and the base plane .
Problem 7: Find
A regular tetrahedron has edge length . Let be the dihedral angle between any two adjacent faces. Find .
Problem 8: find the cosine of the angle between the skew lines and
In a cube with edge length , find the cosine of the angle between the skew lines and .
Problem 9: Find the distance between the two skew lines and
In space, let , , , . Find the distance between the two skew lines and .
Problem 10: Find the distance from the origin to the plane
In space, let , , . Find the distance from the origin to the plane .
Problem 11: Find the distance from to the line
Let the line be . Let . Find the distance from to the line .
Problem 12: Find the distance between the skew lines and
Let be the line through in direction , and let be the line through in direction . Find the distance between the skew lines and .
Problem 13: Find the volume of this circumscribed sphere
A regular triangular pyramid (right pyramid over an equilateral base) has base side length and all three lateral edges equal to . All four vertices lie on a com
Problem 14: Find the surface area of this sphere
A lampshade is a right circular frustum (a truncated cone). The radii of the top and bottom circular rims are and , and the vertical height is . Assume the frus
Problem 15: Find the surface area of that sphere
A regular square frustum has bottom base side length , top base side length , and height . Assume all 8 vertices lie on a common sphere. Find the surface area o
Problem 16: find the radius of the sphere inscribed in the cone (tangent to the base circle and…
A right circular cone is inscribed in a sphere of radius . Among all such cones, the cone volume is maximized. For this maximum-volume cone, find the radius of