Cross Section and Solids of Rotation

A solid is cut by a plane that is parallel to its base, forming a two-dimensional cross section in the shape of a rectangle. Which of the following solids could have resulted in that cross section?

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Select the type of cross section formed when a right hexagonal prism is cut parallel to its base.

• square

• pentagon

• triangle

• hexagon

Answer explanation

When a right hexagonal prism is cut parallel to its base, the cross section formed is identical to the base, which is a hexagon. Therefore, the correct answer is hexagon.

Select the type of cross section formed when a right cone is cut parallel to its base.

• pentagon

• circle

• hexagon

• triangle

Answer explanation

When a right cone is cut parallel to its base, the cross section formed is a circle. This is because the cut maintains the same shape as the base, which is circular.

Select the type of cross section formed when a right cone is cut by a plane containing its altitude and perpendicular to its base.

• pentagon

• triangle

• circle

• rectangle

Answer explanation

When a right cone is cut by a plane that contains its altitude and is perpendicular to its base, the resulting cross section is a triangle. This is because the plane intersects the cone's height and base, forming a triangular shape.

Match the solid below to the two-dimensional shaded region that, upon rotation about the bold axis, would produce the solid. (Note: figures not necessarily drawn to scale.)

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Match the two-dimensional shaded region below to the solid formed by rotating the region about the bold axis. (Note: figures not necessary drawn to scale)

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What shape will the angled cross-section of a rectangular pyramid be?

• Triangle

• Rectangle

• Trapezoid

• Square

Answer explanation

The angled cross-section of a rectangular pyramid intersects the base and the apex, forming a trapezoid. This is because the top edge is narrower than the base, creating two parallel sides.

What shape will the vertical (perpendicular) cross-section of a cube be?

• Square

• Triangle

• Trapezoid

• Circle

Answer explanation

The vertical cross-section of a cube, when cut perpendicularly, will always result in a square shape. This is because all sides of a cube are equal, making any perpendicular slice through the cube a square.

Select all cross sections that can be found in a cylinder

• Circle

• Rectangle

• Semi-Circle

• Triangle

Answer explanation

A cylinder can have circular cross sections when cut parallel to its bases. Rectangular and semi-circular cross sections can occur when cut at an angle or through the side. Triangular cross sections are not possible.

A rectangle and a horizontal line segment are shown. What is the resulting object when the rectangle is rotated around the horizontal line segment?

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