Cross-Sections of Solids
Recall / Warm-Up
Which of the following is a two-dimensional shape?
A rectangular prism has how many faces?
What is a cross-section of a 3D solid?
Fluency Practice
A horizontal plane cuts through a cylinder perpendicular to its axis. What is the shape of the cross-section?
A vertical plane cuts through a cylinder parallel to the cylinder's axis. What is the shape of the cross-section?
A plane cuts through a cone perpendicular to its axis (not through the apex). What is the shape of the cross-section?
A plane cuts diagonally through a rectangular prism, entering one edge and exiting the opposite edge. What is the shape of the cross-section?
A plane cuts through a sphere. What is the shape of the cross-section?
Varied Practice
Which of the following cross-sections is NOT possible when cutting a cylinder?
A vertical cut through a square pyramid passes through the apex and is perpendicular to one base edge. What is the shape of the cross-section?
A plane passes through three non-adjacent vertices of a cube. What is the shape of the cross-section?
A plane cuts through a rectangular prism at a steep angle, passing through parts of four different faces. What is the shape of the cross-section?
A rectangle is rotated 360 degrees around one of its longer sides. Name the 3D solid formed and describe how each dimension of the rectangle relates to a dimension of the solid.
Word Problems / Application
A square pyramid has a base with side length 10 cm. A horizontal plane cuts the pyramid parallel to the base at a point halfway between the base and the apex.
What is the shape and side length of the cross-section?
A right triangle has legs of length 4 cm and 9 cm. The triangle is rotated 360 degrees around the leg of length 9 cm.
What solid is formed, and what are its dimensions?
A semicircle with radius 6 cm is rotated 360 degrees around its diameter.
What 3D solid is formed by this rotation?
What is the radius of the resulting solid?
Error Analysis
Maya was asked: "What cross-section shapes can you get by cutting a cylinder?"
Maya's answer: "The only cross-section of a cylinder is a circle, because the base is a circle."
What is wrong with Maya's reasoning?
David was shown a diagram of a rectangular prism with a cutting plane. He was asked to name the cross-section.
David's answer: "The cross-section is a rectangle, because I can see a rectangular face on the front of the prism."
What mistake did David make?
Challenge / Extension
A right triangle with legs 5 cm and 8 cm is rotated around the leg of length 5 cm. A second copy of the same triangle is rotated around the leg of length 8 cm. How do the two resulting cones compare?
A cone has height 12 cm and base radius 6 cm. A horizontal cross-section is taken 4 cm below the apex. Explain how to find the radius of the cross-section and calculate its area.