Solid Geometry Solutions
40 curated solid geometry problems covering proofs, angles, distances, and comprehensive 3D reasoning with vector methods.
40 problems
Problem 1: Prove that plane
In the quadrilateral pyramid , the base is a rhombus with , . The lateral face is an equilateral triangle, and plane plane . Point is the midpoint of edge . (1)
Problem 2: Prove that
In the triangular frustum , plane plane , is the midpoint of , , and (1) Prove that . (2) When the volume of is , find , where is the angle between line and pla
Problem 3: Prove that points are coplanar
In the right prism , the base is a rhombus with , . Points move on edges , respectively, and satisfy (1) Prove that points are coplanar. (2) If the cosine of th
Problem 4: Prove that plane
In tetrahedron , plane , , , . (1) Prove that plane . (2) If the cosine of the dihedral angle is , find .
Problem 5: Prove that plane
In the quadrilateral pyramid , plane , plane , , and (1) Prove that plane . (2) Let be the reflection of point across plane . Find , where is the angle between
Problem 6: Prove that plane
In the quadrilateral pyramid , , , , , and is equilateral. Points are the midpoints of , respectively. (I) Prove that plane . (II) If the dihedral angle is , fi
Problem 7: Prove that plane
In the quadrilateral pyramid , plane , , , and (1) Prove that plane . (2) Find the cosine of the angle between planes and .
Problem 8: Find the distance from point to plane
In the quadrilateral pyramid , base is a square. Given , and plane . Points are the midpoints of , respectively, and lines , intersect at point . (1) Find the d
Problem 9: Prove that plane
In tetrahedron , . It is given that and . (1) Prove that plane . (2) Find the angle between line and plane .
Problem 10: Prove that , and find the dihedral angle between the two planes
Two lines and intersect at point . Planes and satisfy (Here is a transversal line intersecting .) Prove that , and find the dihedral angle between the two plane
Problem 11: Prove that
In a regular tetrahedron (all edges equal), let and be the midpoints of edges and , respectively. Let and be the midpoints of edges and , respectively. (1) Prov
Problem 12: Prove that
In tetrahedron , it is given that plane . Let be the midpoint of . Define planes (1) Prove that . (2) Find the dihedral angle between and .
Problem 13: find the dihedral angle along edge between faces and
In a unit cube with side length , find the dihedral angle along edge between faces and .
Problem 14: Find the angle between the space diagonal and the base plane
In a right triangular prism , the base is an equilateral triangle with side , and . Find the angle between the space diagonal and the base plane .
Problem 15: Find the dihedral angle along edge between planes and
In a regular square pyramid , the base is a square of side . The apex is directly above the center of the base, and . Find the dihedral angle along edge between
Problem 16: Find the angle between the space diagonal and the base plane
In the right rectangular prism , , , and . Find the angle between the space diagonal and the base plane .
Problem 17: find the distance from vertex to plane
In cube with side length , find the distance from vertex to plane .
Problem 18: Find the distance from vertex to line
Let be a regular tetrahedron with edge length . Let be the centroid of triangle , and let be the median line in the base plane. Find the distance from vertex to
Problem 19: Find the distance between the skew edges and
In a right triangular prism , the base is an equilateral triangle with side , and . Find the distance between the skew edges and .
Problem 20: find the distance from vertex to the face diagonal on the opposite face
In a rectangular box with , , , find the distance from vertex to the face diagonal on the opposite face .
Problem 21: Find the volume of the pyramid
In a right regular hexagonal pyramid , is a regular hexagon with side length . Let be the center of the base and plane with . (1) Find the volume of the pyramid
Problem 22: Find and that maximize the cone’s volume, and give the maximum volume
A right circular cone is inscribed in a sphere of radius . The cone’s axis passes through the sphere center, and its vertex is at the “north pole” of the sphere
Problem 23: Find the ratio
In triangular prism , the top face is a translation of the base, i.e. Plane cuts the prism into two pyramids: tetrahedron and quadrilateral pyramid . Find the r
Problem 24: Find: (1) the slant height;
A right circular frustum has bottom radius , top radius , and height . Find: (1) the slant height; (2) the volume; (3) the total surface area.
Problem 25: 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 26: 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 27: Prove that plane is perpendicular to plane
In tetrahedron , suppose plane and . Prove that plane is perpendicular to plane .
Problem 28: 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 29: 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 30: 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 31: Find
A regular tetrahedron has edge length . Let be the dihedral angle between any two adjacent faces. Find .
Problem 32: 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 33: Find the distance between the two skew lines and
In space, let , , , . Find the distance between the two skew lines and .
Problem 34: Find the distance from the origin to the plane
In space, let , , . Find the distance from the origin to the plane .
Problem 35: Find the distance from to the line
Let the line be . Let . Find the distance from to the line .
Problem 36: 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 37: 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 38: 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 39: 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 40: 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