7th Grade Cross-Sections of Three-Dimensional Figures Lesson (Free PDF)

7th Grade math lesson: Cross-Sections of Three-Dimensional Figures – Maths PDF

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This grade 7 cross sections lesson for 7th Grade covers cross-sections of three-dimensional figures (Common Core 7.G.A.3), with a free PDF to print.

A cross-section is the two-dimensional shape formed when a plane slices a three-dimensional figure. A slice of a rectangular prism parallel to a face is a rectangle the same size as that face. A slice of a rectangular pyramid parallel to its base is a smaller rectangle, and a vertical slice through the apex is a triangle.

Grade 7 · Common Core State Standards: 7.G.A.3

Key facts in this grade 7 cross sections lesson

Key facts

A cross-section is the flat shape where a plane meets a solid.
Prism slice parallel to a face: a rectangle congruent to that face.
Pyramid slice parallel to the base: a smaller rectangle similar to the base.
Pyramid slice through the apex and perpendicular to the base: a triangle.
A slanted plane can give a triangle, trapezoid, or other polygon.

What is a cross-section?

A cross-section is the two-dimensional shape you see when a plane slices through a three-dimensional figure. Imagine cutting a loaf of bread: the cut face is the cross-section.

Cross-section

The cross-section of a solid is the flat shape where a plane meets the solid. Its shape depends on the solid and on how the plane is tilted.

What do slices of a rectangular prism look like?

A right rectangular prism has three pairs of rectangular faces. A plane parallel to any face makes a rectangle that is the same size as that face.

A cut parallel to the base gives a rectangle that is congruent to the base: 8 by 5.

A vertical cut parallel to the front face gives a rectangle 8 by 4.

A slice parallel to the base is congruent to the base, so a prism that is 8 by 5 by 4 gives an 8-by-5 rectangle. A vertical slice parallel to the front face gives an 8-by-4 rectangle, and a vertical slice parallel to the side face gives a 5-by-4 rectangle.

What do slices of a rectangular pyramid look like?

A pyramid narrows toward its apex. A slice parallel to the base is a smaller rectangle with the same shape as the base. A vertical slice that passes through the apex is a triangle.

A cut parallel to the base of a pyramid gives a smaller, similar shape. Halfway up, the sides are half as long: 4 by 4.

A vertical cut through the apex gives a triangle with the same base and height as the pyramid.

For a pyramid of height 12 with a square base of side 8, a cut 6 above the base has \((12 – 6) \div 12 = \frac{1}{2}\) of the base side, so it is a 4-by-4 square with area 16, not 32.

What if the plane is tilted?

A slanted plane can give other shapes. A tilted cut across a cube can make a rectangle that is not a square, or a triangle if it cuts off a corner, or even a hexagon. To name a cross-section, picture where the plane meets each face: each face the plane crosses gives one side of the shape.

Solid	Plane position	Cross-section
Prism	Parallel to a face	Rectangle (same as the face)
Pyramid	Parallel to the base	Smaller similar rectangle
Pyramid	Vertical through the apex	Triangle
Cube	Cuts off a corner	Triangle

Grade 7 cross sections lesson: questions and answers

What is a cross-section?

It is the shape you get when a plane cuts through a solid. For a rectangular prism sliced parallel to the base, it is a rectangle.

What shape do you get when you slice a pyramid parallel to its base?

You get a smaller shape similar to the base. For a square pyramid it is a smaller square.

What shape results from a vertical cut through the apex of a pyramid?

It is a triangle. Its base is a base edge or a line across the base, and its height equals the height of the pyramid.

Can a cube have a triangular cross-section?

Yes. If the plane cuts off a corner by crossing the three edges that meet there, the cut face is a triangle.

How do I find the area of a cross-section?

Identify the shape, find its dimensions from the solid, and use the area formula. For a prism slice parallel to the base, the area equals the base area.

How to use this grade 7 cross sections lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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