Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Slice a loaf of bread straight down and every slice is roughly the same shape. Slice a block of cheese at an angle and the face you reveal is a longer rectangle. These revealed faces are called cross-sections or plane sections, and seventh graders learn to predict them for right rectangular prisms and right rectangular pyramids.
For a prism, a cut parallel to the base gives a rectangle exactly matching the base, while a cut parallel to a side face gives a rectangle matching that face. Tilted cuts can produce larger rectangles, and some cuts through a corner produce triangles. Pyramids behave differently because they narrow toward the apex: a cut parallel to the base gives a smaller rectangle with the same shape as the base, and a vertical cut through the apex gives a triangle. Being able to picture these slices builds spatial reasoning that later supports volume formulas and the study of solids generated by rotation.
Students picture the triangular side faces and assume every cut shows a triangle. Slicing a clay pyramid parallel to its base reveals a rectangle, which surprises many.
Unlike a prism, a pyramid narrows, so a slice closer to the apex is smaller. Halfway up a square pyramid, the square slice has sides half as long as the base.
A flat slice through a solid with flat faces always gives a polygon. A circle appears only when the solid itself has curved surfaces, such as a cylinder or cone.
A right pyramid has a square base with sides of 6 inches. Describe the cross-section from a slice parallel to the base exactly halfway up, and from a vertical slice through the apex.
Answer: The parallel slice is a 3-inch by 3-inch square; the vertical slice through the apex is a triangle.
Physical slicing beats any diagram: clay or modeling dough shaped into prisms and pyramids, cut with dental floss, lets students check predictions instantly. Free 3D geometry apps that show a moving plane through a solid are a good follow-up for students who struggle to visualize.
Tests usually show a solid with a slicing plane and ask students to choose the resulting shape. Encourage students to trace the plane's path across each face it meets, which gives the sides of the cross-section one at a time.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) A rectangle the same size as the base
A prism has the same cross-section all the way up, so a slice parallel to the base is a rectangle identical to the base.
Answer: D) A rectangle smaller than the base with the same shape
The pyramid narrows toward the apex, so a parallel slice is a rectangle similar to the base but smaller.
Answer: C) A triangle
The cut starts at the apex point and spreads to a segment across the base, which outlines a triangle.
Answer: B) A circle
A cube has only flat faces, so a flat slice always makes a polygon. Triangles, rectangles and even hexagons are possible, but a circle is not.
Answer: 40 (also accepted: 40 cm², 40 cm2)
A slice parallel to a face of a prism matches that face: 10 × 4 = 40 cm².
Answer: 4 (also accepted: 4 cm)
Halfway to the apex, the square has shrunk to half the base side: 8 ÷ 2 = 4 cm.
A full lesson with slides, activities and an exit ticket on cross-sections of 3d figures, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.G.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn cross-sections of 3d figures into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It is the flat shape you see where a plane (a flat cut) meets a solid. Plane section and cross-section mean the same thing in this standard.
The standard names right rectangular prisms and right rectangular pyramids. Many teachers extend the idea to cylinders and cones as enrichment.