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Exercises: Identify Cross-Sections and Solids of Revolution

Grade 10·20 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-gmd-b-4
Work through problems with immediate feedback
A

Warm-Up

1.

A cross-section is best described as:

2.

A horizontal slice (parallel to the base) is taken through a cone. What shape is the cross-section?

3.

A solid of revolution is created by:

B

Fluency Practice

A cylinder with a vertical plane through its axis, highlighting the cut region and asking what shape the cross-section is
1.

A vertical plane passes through the axis of a cylinder (a plane containing the cylinder's central axis). What shape is the cross-section?

2.

A sphere of radius RR is sliced by a plane at a distance dd from the center, where 0<d<R0 < d < R. What shape is the cross-section, and how does its size compare to a cross-section through the center?

A cylinder with an oblique cutting plane highlighting the cut region and asking what shape the cross-section is
3.

A cylinder is sliced by a plane that is tilted at an angle to the axis (oblique to the axis, not perpendicular and not parallel). What shape is the cross-section?

4.

A right triangle is rotated 360°360\degree about its vertical leg. Which solid is produced?

5.

A rectangle with width 3 cm and height 8 cm is rotated about its longer edge (height = 8 cm). Which solid is produced, and what are its dimensions?

C

Varied Practice

1.

For each solid, identify the cross-section shape produced by the described cut.

  • A rectangular prism cut by a plane parallel to a rectangular face:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
  • A square pyramid cut by a horizontal plane (parallel to the base, halfway up):   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
  • A sphere cut by any plane:   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  
rectangular prism cross-section:
pyramid cross-section:
sphere cross-section:
A cone with a vertical plane through the apex, highlighting the cut region and asking what shape the cross-section is
2.

A cone is sliced by a vertical plane that passes through the apex (the tip) of the cone. What shape is the cross-section?

A rectangle rotated about its short edge and long edge, showing two different cylinder shapes with unknown dimensions
3.

The same rectangle (width 4 cm, height 10 cm) is rotated about two different axes. Rotation 1: about the short edge (4 cm). Rotation 2: about the long edge (10 cm). Which statement correctly compares the two solids?

Four 2D shapes beside a dashed axis: quarter-circle (A), semicircle with axis on diameter (B), right triangle (C), rectangle (D)
4.

Which two-dimensional shape, when rotated 360°360\degree about the dashed axis shown, produces a sphere?

D

Word Problems

A sphere of radius 6 m showing a center cut producing a great circle of radius 6 m and an off-center cut at y = 4 m
1.

A water tower tank has the shape of a sphere with radius 6 meters. An engineer needs to analyze the cross-sections to design internal supports.

1.

The engineer takes a horizontal cross-section through the center of the sphere. What is the radius of this cross-section circle?

2.

The engineer also takes a horizontal cross-section at a height of 4 m above the center of the sphere. Using rslice=R2−d2r_{\text{slice}} = \sqrt{R^2 - d^2} where R=6R = 6 m is the sphere's radius and d=4d = 4 m is the distance from the center, find the radius of this cross-section circle. Round to the nearest tenth.

A right trapezoid with its perpendicular side on the rotation axis, and the resulting 3D solid produced by rotating it 360 degrees, labeled 'What solid is this?'
2.

A manufacturing company uses a lathe to produce metal parts by spinning a 2D profile about an axis. The engineer designs each part by specifying the 2D profile and the axis of rotation.

1.

The engineer rotates a right trapezoid (shown in the diagram) 360°360\degree about its perpendicular side (the vertical leg). Which solid is produced?

2.

The engineer wants to produce a part whose cross-section (taken perpendicular to the axis) at any height is always a circle. Is this guaranteed for every solid of revolution? Explain why or why not.

E

Error Analysis

Two cylinders side by side: one with a horizontal cut and one with an oblique cut, prompting the student to identify the oblique cross-section shape
1.

A teacher asks: "A cylinder is sliced by a plane that is not perpendicular to the axis and not parallel to the axis — it is tilted at 30° to the horizontal. What shape is the cross-section?"

Jaylen answers: "It's a circle. Cylinders always have circular cross-sections because the cylinder itself is circular."

What is wrong with Jaylen's reasoning? Select the best correction.

2.

A student is asked: "What solid is produced when a rectangle is rotated about one of its edges?"

The student writes: "You always get the same cylinder no matter which edge you rotate around — the shape is a rectangle either way, so the solid is always the same cylinder."

Which statement best identifies and corrects the student's error?

F

Challenge / Extension

1.

A cube has side length ss. A plane is passed through the midpoints of six edges — two edges on each of three pairs of opposite faces — so that it cuts through all six faces of the cube. What shape is the cross-section? Explain your reasoning, including why the cross-section cannot be a rectangle or triangle.

2.

A solid of revolution is formed by rotating a right trapezoid about its perpendicular side. The trapezoid has: perpendicular side (axis) of length 6 cm, bottom horizontal edge of length 5 cm, and top horizontal edge of length 2 cm. The resulting solid is a frustum (truncated cone). Using Vfrustum=13πh(R2+Rr+r2)V_{\text{frustum}} = \frac{1}{3}\pi h (R^2 + Rr + r^2) where h=6h = 6 cm, R=5R = 5 cm, and r=2r = 2 cm, compute the volume in terms of π\pi.

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