The Cross-Section Made by Cutting a Cube Straight Across with a Plane
Let us consider what shape can be seen when a cube is cut straight across by a flat surface (a plane).
Various Cross-Sections of a Cube
- Cut through 3 vertices → equilateral triangle
- Cut through the midpoints of 3 edges → equilateral triangle or regular hexagon
- Cut so that it includes a diagonal → rectangle or square
- Cut through the midpoints of 6 edges → regular hexagon
Example 1: A cube with side length 6 cm is cut by a plane passing through 3 vertices. What are the shape and area of the cross-section?
Solution:
① The cross-section is an equilateral triangle
② The side length is the face diagonal, 6√2 cm
③ Area = (√3 / 4) × (6√2)² = (√3 / 4) × 72 = 18√3 cm²
Solution:
① The cross-section is an equilateral triangle
② The side length is the face diagonal, 6√2 cm
③ Area = (√3 / 4) × (6√2)² = (√3 / 4) × 72 = 18√3 cm²
Example 2: A cube with side length 4 cm is cut by a diagonal plane passing through opposite edges. What are the shape and area of the cross-section?（√2 ≒ 1.41）
Solution:
① The cross-section is a rectangle（length 4 cm × width 4√2 cm）
② Area = 4 × 4 × 1.41 ≒ 22.56 cm²
Solution:
① The cross-section is a rectangle（length 4 cm × width 4√2 cm）
② Area = 4 × 4 × 1.41 ≒ 22.56 cm²
Rule for cross-section problems: “Two straight lines on the same plane either intersect or are parallel.” Using this rule helps us determine the shape of the cross-section.