The two-dimensional shape that is revealed when a three-dimensional solid is sliced through by a flat plane.
Slice an orange—the cut surface is a cross-section (a circle).
Showing a random 20 of 50 problems.
Example 1
easy
A cylinder of radius 5 cm and height 12 cm is cut by a horizontal plane halfway up its height. Describe and find the area of the cross-section.Cylinder cut by a horizontal plane halfway up its height
Example 2
easy
What is the cross-section of a sphere sliced by a plane tangent to it?
Example 3
challenge
A cube of side 1 is sliced by a plane through the midpoints of six edges, forming a regular hexagonal cross-section. Find the area of that hexagon.
Example 4
easy
A rectangular box is sliced parallel to one face. What is the cross-section?
Example 5
easy
A cone is sliced straight down through its apex. What shape appears?
Example 6
medium
A rectangular prism is 3×5×7 cm. What is the largest possible area of a planar cross-section parallel to one of its faces?Rectangular prism 3 × 5 × 7 cm — which parallel face cut gives the largest cross-section?
Example 7
medium
A sphere of radius 5 is sliced 3 units from its center. What is the radius of the circular cross-section?Sphere of radius 5 sliced 3 units from its center — find the cross-section radius
Example 8
challenge
Explain why a plane can slice a cube into a regular hexagon, naming the maximum number of sides a cube cross-section can have.
Example 9
medium
A cube of side 6 is sliced by a plane perpendicular to a face diagonal passing through the cube's center. Describe the cross-section shape.
Example 10
hard
A sphere of radius r is cut by two parallel planes equidistant from the center, each at distance d from it. Find the ratio of the cross-section areas.
Example 11
easy
True or false: a cylinder's cross-section parallel to the base is a circle.
Example 12
hard
A cone with base radius 6 cm and height 12 cm is cut by a plane parallel to the base at height 8 cm from the base. Find the radius and area of the cross-section, then compute the ratio of areas (cross-section : base).Cone with base radius 6 cm and height 12 cm; horizontal cut at height 8 cm
Example 13
medium
A cube of side 4 is sliced by a plane through three edges, cutting each at the midpoint. What is the cross-section shape?
Example 14
medium
A square pyramid with a 6 cm ×6 cm base and height 9 cm is cut by a horizontal plane 3 cm above the base. Find the shape and dimensions of the cross-section.
Example 15
hard
A cone has a base radius of 6 cm. A plane perpendicular to the base passes through the center, cutting the cone into two equal pieces. If the cone's height is 9 cm, find the area of this cross-section.Cone with radius 6 cm and height 9 cm; vertical cross-section through the axis
Example 16
medium
A square-based pyramid is sliced parallel to its base. What is the cross-section?
Example 17
easy
A cube is sliced parallel to one of its faces. What is the cross-section?
Example 18
hard
A cylinder of radius 5 and height 20 is cut by a plane tilted at 45° to its axis passing through the center. What shape is the cross-section?
Example 19
medium
Why does slicing a cone give such different shapes (circle, ellipse, parabola)?
Example 20
challenge
A sphere of radius 13 is sliced, and the circular cross-section has radius 12. How far from the center was the slice made?Sphere of radius 13; cross-section circle has radius 12 — find the distance from the center