Cross-Section Math Example 3

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Example 3

medium
A cone is cut by a plane parallel to its base. What shape is the cross-section?

Solution

  1. 1
    A cone has a circular base and tapers to a point (apex). Any plane parallel to the base intersects the lateral surface in a curve similar to the base.
  2. 2
    Since the base is a circle and the plane is parallel to it, the cross-section is also a circle by the property of similar cross-sections in cones.
  3. 3
    The radius of the cross-section is smaller than the base radius. If the plane is at height hh from the base and the cone has total height HH and base radius RR, then the cross-section radius is r=R⋅H−hHr = R \cdot \frac{H - h}{H}.

Answer

A circle (smaller than the base)\text{A circle (smaller than the base)}
When a plane cuts a cone parallel to its base, the resulting cross-section is always a circle. This follows from the rotational symmetry of the cone. The closer the cut is to the apex, the smaller the circle becomes.

About Cross-Section

The two-dimensional shape that is revealed when a three-dimensional solid is sliced through by a flat plane.

Learn more about Cross-Section →

More Cross-Section Examples

Example 1 easy

A cylinder of radius [formula] cm and height [formula] cm is cut by a horizontal plane halfway up it

Example 2 medium

A square pyramid with a [formula] cm [formula] [formula] cm base and height [formula] cm is cut by a

Example 4 easy

What shape is the cross-section when a sphere is cut by any plane through its centre? What is the ar

Example 5 hard

A cone with base radius [formula] cm and height [formula] cm is cut by a plane parallel to the base

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