7th Grade · 7.G.A.3

Cross sections of 3D figures in 7th grade

Cross sections of 3D figures are the flat shapes you see when you slice a solid with one straight cut. A 3D figure is a solid shape, such as a box or a pyramid. In 7th grade, your child learns to picture the slice and name the 2D shape it makes, such as a rectangle, a triangle or a trapezoid.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

What it means, in plain words

Picture a loaf of bread. Each slice has a flat face, and the shape of that face is the cross section. In math, the cut is a plane, a perfectly flat surface like a sheet of glass that goes straight through the solid. The cross section is the 2D shape where the plane meets the solid.

In 7th grade, slicing 3D figures starts with two solids. A right rectangular prism is a box shape, like a cereal box, with six rectangle faces that meet at square corners. A right rectangular pyramid has a rectangle for a base and four triangle sides that rise to one point at the top. A cut parallel to the base (level with it) makes a copy of the base's shape. In a box that copy is the same size at every height. In a pyramid it gets smaller as the cut moves up. A cut straight down through the top point of a pyramid makes a triangle. Some other cuts make a trapezoid (a four-sided shape with one pair of parallel sides).

One check works for all cross sections of 3D figures with flat faces: count the faces the cut passes through. Each face adds one straight side to the slice. A cut through three faces gives a triangle, and a cut through four faces gives a four-sided shape.

Your child should be able to:

Name the 2D shape made by a cut parallel to the base of a box or a pyramid
Describe the slice made by a cut straight down or at a slant
Explain why a level slice of a pyramid is smaller than its base
Count the faces a cut passes through to find how many sides the slice has

Example

A box is 10 inches long, 4 inches wide and 6 inches tall. A flat cut goes through it parallel to the bottom. What shape is the cross section, and how big is it?

A cut parallel to the bottom gives a copy of the bottom face.
The bottom is a rectangle 10 inches long and 4 inches wide.
The sides of a box go straight up, so the slice is the same size at every height.
The cross section is a rectangle 10 inches long and 4 inches wide.

Example

A flat cut slices one corner off a cube. What shape is the cross section?

Three faces of the cube meet at that corner.
The cut passes through those three faces and no others.
Each face adds one straight side to the slice.
The cross section has three sides, so it is a triangle.

Where children usually get stuck

Asked for the cross section of a box, your child answers "a prism" or draws a smaller box.
The child mixes up the piece that was cut off with the flat face the cut leaves behind. The cross section is only that flat face, so it is always a 2D shape.
What helps. Cut a block of cheese or a kitchen sponge and press the cut face onto paper like a stamp. The outline on the paper is the cross section.
Your child says a slice of a box can only be a rectangle, because every face of a box is a rectangle.
The faces are what the child can see on the solid. A slanted cut that passes through three faces gives a triangle, and a cut through five faces gives a five-sided shape.
What helps. Before naming the shape, ask your child to count how many faces the cut passes through. That count is the number of sides of the slice.
On a drawing of a slanted cut, your child counts too few sides because some edges of the slice are at the back.
A drawing of a solid is flat, and the back faces are drawn with dashed lines or not at all. The child counts only the edges in front.
What helps. Hold a real box, such as a tissue box, and trace the path of the cut across each face with a finger. Then count the faces the finger touched.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
A flat cut goes through a pyramid with a rectangular base that is not a square. The cut is level: it is parallel to the base. What 2D shape is the slice?
A
triangle
B
rectangle
C
pentagon
D
hexagon
Show the answer and how to solve it
The slice is a rectangle that is not a square, smaller than the base. A level cut copies the base, and this base is a rectangle that is not a square.
1. 1The outline shows the slice, with a dot at each corner. The cut is level: it is parallel to the base and passes below the top point. It crosses all 4 side faces, so the slice has 4 sides. The side on the back face is dashed because that face is hidden.
2. 2On each side face, the cut runs parallel to the bottom edge of that face. So each corner of the slice matches a corner of the base: all 4 are square corners. The marks show two of them.
3. 3The pyramid gets narrower evenly toward its top point, so a level cut makes a smaller copy of the base. The base is a rectangle that is not a square, so the slice is also a rectangle that is not a square: longer than it is wide, like the base, but smaller.
Common mistake. A common mistake is to pick a triangle because the side faces of a pyramid are triangles. The slice is not a side face: the level cut crosses every side face and makes one side on each, so the slice has as many sides as the base.
The idea underneath. A level cut that passes below the top point of a pyramid makes a smaller copy of the base. So the slice has the same shape as the base: name the base and you have named the slice.
2On level
Each diagram shows a solid cut by a flat plane. Match each diagram to the shape of its slice. One shape card is extra and will not be used.
A: a rectangular prism with no square faces
B: a pyramid with a square base
C: a pyramid with a rectangular base that is not a square
Connect each item on the left to its match on the right.
Diagram
2D shape
A
B
C
Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.
Show the answer and how to solve it
A matches the trapezoid, B matches the square and C matches the triangle. The pentagon card is not used.
1. 1In each diagram, the see-through sheet is the plane. It cuts the solid into a green piece and a pale piece. The slice is the flat shape where the plane cuts through the solid. In the pictures below, the slice is outlined, and a dashed side is one you cannot see from the front. There are 3 diagrams and 4 shape cards, so one card will be left over.
2. 2Diagram A shows a rectangular prism. The plane slants and cuts a wedge off the front right tall edge, the edge nearest you. It crosses four faces: the top, the front, the right and the bottom face, so the slice has four sides. The top and bottom faces are parallel, so the sides on them are parallel. The plane is closer to that edge at the top than at the bottom, so the top side is shorter than the bottom side. Two parallel sides of different lengths mean the other two sides are not parallel. A four-sided shape with exactly one pair of parallel sides is a trapezoid.
3. 3Diagram B shows a pyramid with a square base. The plane is level, halfway up, parallel to the base. A level cut of a pyramid makes a smaller copy of its base. The base is a square, so the slice is a square.
4. 4Diagram C shows a pyramid with a rectangular base. The plane stands straight up and passes through the top point. It crosses the left face, the right face and the base. The slice has one side on each of these faces: two sides run down from the top point, and the third side runs across the base. So the slice is a triangle.
5. 5The pentagon card is the extra card. No slice here has five sides.
Common mistake. A common mistake is matching diagram C to the trapezoid card. Its plane passes through the top point, so two sides of the slice meet there and the slice has three sides.
The idea underneath. When a plane cuts a pyramid level with its base, the slice is a smaller copy of the base. The slice has one side on each face the plane crosses, so count the faces it crosses. Where the plane crosses two parallel faces, the two sides it makes are parallel.
3On level
Look at the slice in the pyramid. Match each claim to the reasoning or mistake it shows.
A plane cuts a rectangular pyramid parallel to the base.
Connect each item on the left to its match on the right.
Claims
Reasoning or mistake
The slice is a triangle because the cut passes through the tip.
The cross-section is a rectangle.
The slice is a nonrectangular parallelogram because it looks slanted.
Mistook the perspective drawing for the actual flat cross-section
Correct: a slice parallel to the base keeps the base shape
Assumed the cut passes through the apex of the pyramid
Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.
Show the answer and how to solve it
Match "The slice is a triangle because the cut passes through the tip." to "Assumed the cut passes through the apex of the pyramid". Match "The cross-section is a rectangle." to "Correct: a slice parallel to the base keeps the base shape". Match "The slice is a nonrectangular parallelogram because it looks slanted." to "Mistook the perspective drawing for the actual flat cross-section".
1. 1The plane cuts the rectangular pyramid parallel to its base and below its tip. The slice is smaller than the base.
2. 2The cut is parallel to the rectangular base. Each side of the slice is parallel to the corresponding base side, so the slice is a smaller rectangle with four right angles. Match "The cross-section is a rectangle." to "Correct: a slice parallel to the base keeps the base shape".
3. 3This claim assumes the cut passes through the tip, also called the apex. The shown cut stays below the tip and is parallel to the base, so that assumption is wrong. Other kinds of cuts can make triangles; a triangle does not always require a cut through the tip. Match "The slice is a triangle because the cut passes through the tip." to "Assumed the cut passes through the apex of the pyramid".
4. 4The drawing shows a 3D view, which can make right angles look slanted. The real slice has four right angles. A rectangle is a parallelogram, but it is not a nonrectangular parallelogram. Match "The slice is a nonrectangular parallelogram because it looks slanted." to "Mistook the perspective drawing for the actual flat cross-section".
Common mistake. Check the stated assumption in each claim. This cut is parallel to the base, so the slice has four right angles. A tilted plane and a slanted drawing are different ideas.
The idea underneath. When a pyramid is sliced by a plane parallel to its base, the cross-section has the same shape as the base, even if the drawing makes it look slanted.
4Stretch
These slices were sorted by the shape of each slice.
The cards show a square pyramid and a rectangular pyramid (its base is a rectangle that is not a square), each cut by a flat plane.
Two cards are in the wrong bins. Drag each one to the correct bin.
Drag each item into the group where it belongs, or tap the item, then tap the group.
Triangle cross section
Trapezoid cross section
Keyboard instructions: focus an item and press Enter or Space to pick it up. Move focus to a bin or the bank and press Enter or Space to drop it. Press Escape to cancel.
Show the answer and how to solve it
Move the square pyramid with the cut that rises from the front base edge to Trapezoid cross section and the square pyramid with the corner cut to Triangle cross section. Every other card is already in the bin that matches its slice.
1. 1Check each card. Find where the plane meets the faces of the pyramid. Each face the plane cuts across gives the slice one side.
2. 2The classmate put this rectangular pyramid in Triangle cross section. This plane stands upright and passes through the top point. It cuts across the left face, the right face and the base, so the slice has 3 sides. The slice is a triangle. It stays.
3. 3The classmate put this square pyramid in Triangle cross section. This plane is slanted and passes below the top point. It runs along the whole front base edge, so that edge is one side of the slice. Then it cuts across the left face, the back face and the right face, so the slice has 4 sides. The front base edge and the side on the back face are parallel. The other two sides lean toward each other. The slice is a trapezoid. Move it to Trapezoid cross section.
4. 4The classmate put this square pyramid in Trapezoid cross section. This plane is slanted and passes below the top point. It cuts across the front face, the left face, the right face and the base, so the slice has 4 sides. The side on the base and the side on the front face both run the same way as the front base edge, so they are parallel. The other two sides lean toward each other. The slice is a trapezoid. It stays.
5. 5The classmate put this square pyramid in Trapezoid cross section. This plane is slanted and misses the top point. It cuts off one corner of the base, so it crosses only the front face, the right face and the base. The slice has 3 sides, so it is a triangle. Move it to Triangle cross section.
6. 6Move the square pyramid with the cut that rises from the front base edge to Trapezoid cross section and the square pyramid with the corner cut to Triangle cross section. The other cards stay where they are.
Common mistake. A common mistake is to name the shape of the front face instead of the slice. When the plane rises from the front base edge, the whole front face stays in front of the plane and looks like a triangle. The slice is the flat face along the plane, and it has 4 sides.
The idea underneath. Find where the plane meets the faces of the pyramid and count the sides of the slice. A slanted cut that misses the top point can still make a triangle, so check every card.