Cross-Section Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
A square pyramid with a 66 cm ×\times 66 cm base and height 99 cm is cut by a horizontal plane 33 cm above the base. Find the shape and dimensions of the cross-section.

Solution

  1. 1
    Step 1: The cross-section of a pyramid cut by a horizontal plane is always a square (similar to the base, by similar triangles).
  2. 2
    Step 2: The cut is at height h=3h = 3 cm from the base, out of total height H=9H = 9 cm. The fraction from the apex is (9−3)/9=6/9=2/3(9-3)/9 = 6/9 = 2/3.
  3. 3
    Step 3: The cross-section side length =23×6=4= \dfrac{2}{3} \times 6 = 4 cm (scales with the ratio from the apex).
  4. 4
    Step 4: Area =42=16= 4^2 = 16 cm2^2.

Answer

Square cross-section with side 44 cm; area =16= 16 cm2^2.
Horizontal cross-sections of a pyramid are squares similar to the base. By similar triangles, the ratio of cross-section side to base side equals the ratio of the remaining height (apex to cut) to the full height.

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 3 medium

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

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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