Volume after Cutting
When a cube is cut diagonally, let us find the volume of the smaller part that is cut off. The key is to think of it as a triangular pyramid or a quadrangular pyramid.
How to Find the Volume after Cutting
- Try dividing the cut-off solid into triangular pyramids or quadrangular pyramids
- Volume of a triangular pyramid = (1/3) × base area × height
- Sometimes it is easier to find the answer by using “larger solid − removed pyramid”
Example 1: A cube with side length 6 cm is cut by a plane passing through vertex A and the 3 adjacent vertices. Find the volume of the triangular pyramid that is cut off.
Solution:
① The part cut off is the triangular pyramid formed by A and the 3 adjacent vertices
② The base is a right triangle → (1/2) × 6 × 6 = 18 cm², height = 6 cm
③ Volume = (1/3) × 18 × 6 = 36 cm³（1/6 of the entire cube, which is 216 cm³）
Solution:
① The part cut off is the triangular pyramid formed by A and the 3 adjacent vertices
② The base is a right triangle → (1/2) × 6 × 6 = 18 cm², height = 6 cm
③ Volume = (1/3) × 18 × 6 = 36 cm³（1/6 of the entire cube, which is 216 cm³）
Example 2: A rectangular prism measuring 4 cm in length, 6 cm in width, and 3 cm in height is cut by a vertical plane passing through the diagonal of the base. Find the volume of the smaller triangular prism.
Solution:
① The base is a right triangle formed by dividing a 4 cm × 6 cm rectangle in half
② Base area = (1/2) × 4 × 6 = 12 cm²
③ Volume = 12 × 3 = 36 cm³
Solution:
① The base is a right triangle formed by dividing a 4 cm × 6 cm rectangle in half
② Base area = (1/2) × 4 × 6 = 12 cm²
③ Volume = 12 × 3 = 36 cm³