Swim Practice: Solid Slices

Describe the cross sections of models used on a team's practice display.

Lap Models

The swim team uses solid foam models on its practice display. A cross section is the flat shape where a plane cuts through a solid. Name the most specific shape possible. All cuts pass through the interior.

A right rectangular prism represents a 24-lap practice set. It stands on an 8 cm by 5 cm rectangular base and is 12 cm tall. Its top and bottom are parallel.

A plane cuts the lap model halfway up, parallel to its base. Which describes the cross section?

A. A square with side length 5 cm.B. A triangle with a base of 8 cm.C. A rectangle measuring 8 cm by 5 cm.D. A rectangle measuring 8 cm by 12 cm.

A second plane cuts the same model 3 cm above the bottom, parallel to the base.

What is the length of the longer side of this cross section? cm

For a lane-assignment card, Nia needs a cross section measuring 5 cm by 12 cm. Which cut produces it?

A. A cut parallel to a 5 cm by 12 cm face.B. A cut removing a small triangular corner.C. A cut parallel to an 8 cm by 12 cm face.D. A cut parallel to the 8 cm by 5 cm base.

A vertical plane contains a diagonal of the model's bottom face and the matching diagonal of its top face. The plane is perpendicular to the base.

What shape is the cross section?

Practice-Time Pyramids

Use the pyramid description for Questions 5–9. Imagine the plane's intersection with the solid, not the shape of a leftover piece.

The team marks its 40-minute practice goal with a right square pyramid. Its base is a 10 cm square, its height is 12 cm, and its apex is directly above the center of the base.

A horizontal plane cuts the pyramid 4 cm above its base, parallel to the base.

What shape is the cross section?

Two planes cut the pyramid parallel to its base: one at a height of 3 cm and one at a height of 9 cm. How do the cross sections compare?

A. Both are squares; the higher one is larger.B. Both are squares; the higher one is smaller.C. Both are squares with equal side lengths.D. The lower one is square; the higher one is triangular.

A vertical plane passes through the apex and the midpoints of two opposite sides of the square base.

What shape is the cross section?

Another vertical plane passes through the apex and two opposite corners of the square base.

How many sides does this cross section have?

Luis wants a smaller square template for the team's practice-time cards. Which slicing plan will produce one?

A. Cut vertically through the apex and opposite side midpoints.B. Cut vertically through the apex and a base diagonal.C. Cut along a triangular face without entering the interior.D. Cut through the interior parallel to the square base.

Check the Slice Plans

Apply what you know about slicing planes. Count only the sides of the flat cross section.

10.

The team uses a cube-shaped model for an 8-lap warm-up. A plane clips off one corner, crossing each of the three edges that meet at that corner and no other edges. What is the cross section?

A. The cross section is a triangle.B. The cross section is a pentagon.C. The cross section is a square.D. The cross section is a hexagon.

11.

For a new practice badge, a plane slices a cube through the interiors of all six faces. Each face contributes one side of the cross section.

What is the name of this six-sided cross section?

12.

Match each practice-display slice to its cross section. Use each description once.

13.

Amara says, “If a cut is parallel to the base, its cross section must be the same size as the base.” Compare the lap-model prism and the practice-time pyramid. Explain why her statement works for one model but not the other.