A cross-section is the flat, two-dimensional shape revealed when a plane cuts through a three-dimensional solid. For example, slicing a cylinder parallel to its base gives a circle, while slicing it at an angle gives an ellipse.
Imagine slicing a loaf of breadβeach slice reveals a 2D shape. The shape you see depends on the angle and position of your cut. Slice a cylinder straight across and you get a circle; slice it at an angle and you get an ellipse. Slice a rectangular prism and you can get rectangles, triangles, or even hexagons depending on the cut.
Showing a random 20 of 50 problems.
Example 1
medium
Slicing a square pyramid parallel to its base yields what shape?
Example 2
medium
A right circular cone has base radius 6 and height 9. A plane parallel to the base cuts the cone at a height 3 above the base. Find the area of the cross-section.Cone (r = 6, h = 9) β horizontal cut at height 3 above the base; find the cross-section area
Example 3
easy
Slicing a cube parallel to a face gives what shape?
Example 4
easy
What 2D shape do you get from a horizontal cross-section of a square prism?
Example 5
easy
What determines the shape of a cross-section?
Example 6
hard
What 2D shape results from cutting a regular octahedron by a horizontal plane through its center?
Example 7
easy
Which solid has a circular cross-section no matter how it is sliced (as long as the slice is a flat plane)?
Example 8
hard
A cube of edge length 2 is cut by a plane that passes through the midpoints of six edges, forming a regular hexagonal cross-section. What is the side length of the hexagon?Cube of edge length 2 β hexagonal cross-section through midpoints of six edges
Example 9
easy
A plane cuts through a sphere. What shape is the cross-section, and when is it the largest possible cross-section?
Example 10
challenge
Explain why a CT scan reconstructs a 3D body from many 2D cross-sections.
Example 11
medium
A cone (base radius 6, height 12) is sliced parallel to the base at height 8 (measured from the base). Find the cross-section radius.Cone (r = 6, h = 12) β horizontal cut at height 8 above the base
Example 12
challenge
A cube of side 6 is cut by a plane through three vertices that meet at one corner. What shape and what type of triangle results?
Example 13
medium
A cone of base radius 6 and height 12 is sliced parallel to the base, halfway up. What is the radius of the circular cross-section?Cone with base radius 6 and height 12 β sliced parallel to the base at the midpoint
Example 14
medium
A sphere has radius 13. A horizontal plane cuts the sphere 5 units above the center. What is the radius of the cross-section?Sphere of radius 13 β plane cuts 5 units above the center; find the cross-section radius
Example 15
easy
What shape is a horizontal cross-section of a cone (parallel to base)?
Example 16
medium
Which conic sections come from slicing a double cone at different angles?
Example 17
easy
What 2D shape is produced by a vertical cross-section of a cylinder taken parallel to its axis and passing through the center?
Example 18
easy
What is the cross-section of a triangular prism cut parallel to its triangular base?
Example 19
challenge
A unit cube is cut by a plane through three midpoints of edges that all meet at one vertex (i.e., the midpoints of the three edges emanating from a single corner). What is the area of the cross-section?
Example 20
hard
A cone has base radius 9 and height 12. A horizontal cross-section has area 4Ο. How far above the base is the cutting plane?Cone (r = 9, h = 12) β horizontal cross-section has area 4Ο; find the height of the cut above the base