7.G.A.3: Slicing Prisms and Pyramids (Plane Sections)
In plain English: 7.G.A.3 is the Common Core grade 7 math standard that asks students to describe the flat shapes made when a solid is sliced by a plane (a flat surface). It focuses on boxes (right rectangular prisms) and right rectangular pyramids: students name each slice, such as a rectangle, triangle, trapezoid (one pair of parallel sides), pentagon or hexagon, and explain its size and why some shapes are impossible.
Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them. Also written as 7.G.3 · Official standard
Students imagine, draw and cut slices of two kinds of solids. A right rectangular prism is a box: it has 6 rectangle faces, and its side faces stand straight up from its base. A right rectangular pyramid has a rectangle base and 4 triangle faces that meet at one point, the apex, which sits directly above the center of the base. A face is a flat surface of a solid, and an edge is a segment where two faces meet.
A plane is a flat surface that goes on forever in every direction, like a sheet of paper with no edges. When a plane slices a solid, the flat shape left on the cut surface is called a plane section or cross section. Students describe each section by its shape, its number of sides, which sides are parallel (always the same distance apart, like railroad tracks), and its measurements when they can find them. A key idea: each side of the section lies on a different face of the solid, so a box (6 faces) can give at most a six-sided section and a pyramid (5 faces) at most a five-sided one.
Learning Objectives
By the end of this lesson, students will be able to:
Describe the slices of a box made by planes parallel to its faces, including their dimensions
Show how tilted slices of a box give triangles, four-sided shapes, pentagons and hexagons, and explain why seven sides is impossible
Describe slices of a right rectangular pyramid parallel to the base as smaller rectangles of the same shape, and find their dimensions halfway up
Describe the triangles and trapezoids made by slicing a pyramid straight down, and explain why a pyramid has no six-sided slice
Prior Knowledge Required
Students should already be comfortable with:
The faces, edges and vertices of prisms and pyramids, and their nets 6.G.A.4
The area of rectangles, triangles and trapezoids 6.G.A.1
Parallel and perpendicular lines 4.G.A.1
Naming polygons by their number of sides: triangle, quadrilateral (4 sides), pentagon (5), hexagon (6) 5.G.B.4
Hold up a loaf of sliced bread or a picture of one, and a block of cheese. Ask students to answer the prompt on paper with a quick sketch.
Warm-Up Prompt
"A block of cheese is shaped like a box. You cut straight across it, parallel to the end. What shape is the cut face? Could you make a cut whose face is a triangle? Sketch your idea."
Most students say the straight cut gives a rectangle, and it does. Collect a few ideas for the triangle. If no one has one, hint: "What if you cut off just one corner?" Tell students that today they will predict, sketch and check every kind of slice of a box and of a pyramid.
Direct Instruction20 minutes
Show Diagram 1. The box is a block of cheese 10 cm long, 6 cm wide and 4 cm tall. Remind students of the vocabulary: the base is the face the solid rests on, a slice parallel to the base is level with the table, and a slice straight down is perpendicular to the table (at a right angle to it). Work through the examples, sketching each slice on the board.
Box, slice parallel to the base
The block of cheese is 10 cm long, 6 cm wide and 4 cm tall. Slice it parallel to the base, 2 cm up.
Equation: The slice is a rectangle 10 cm by 6 cm, exactly the size of the base, with an area of 10 × 6 = 60 cm². Every slice parallel to the base is this same rectangle, at any height (Diagram 1, A).
Box, slices straight down
Slice the same block straight down, first parallel to the long 10 cm by 4 cm front face, then parallel to the 6 cm by 4 cm end face.
Equation: Parallel to the front face: a rectangle 10 cm by 4 cm. Parallel to the end face: a rectangle 6 cm by 4 cm (Diagram 1, B). A slice parallel to any face is a copy of that face.
Box, tilted slices
Tilt the knife. What shapes can a tilted slice of the block make?
Equation: Cutting off one corner crosses 3 faces and makes a triangle (Diagram 1, C). A tilted cut that crosses 4 faces makes a four-sided shape, 5 faces a pentagon, and all 6 faces a hexagon (Diagram 1, D). Each side of the slice lies on a different face, and a box has only 6 faces, so no slice has 7 or more sides.
Pyramid, slices parallel to the base
A candle is a right rectangular pyramid with an 8 cm by 6 cm base, 6 cm tall. Slice it parallel to the base halfway up, and then three quarters of the way up.
Equation: Each slice is a rectangle with the same shape as the base, but smaller. Halfway up (3 cm), each side is half as long: 4 cm by 3 cm (Diagram 2, A). Three quarters of the way up (4.5 cm), each side is a quarter as long: 2 cm by 1.5 cm. The slices shrink to a single point at the apex.
Pyramid, slices straight down
Slice the candle straight down, parallel to the 8 cm edge: once through the apex, and once 1.5 cm in front of the center.
Equation: Through the apex: a triangle with an 8 cm base and a height of 6 cm, the height of the pyramid, so its area is ½ × 8 × 6 = 24 cm² (Diagram 2, B). In front of the center: a trapezoid, a four-sided shape with exactly one pair of parallel sides. Its sides are 8 cm along the base and 4 cm at the top, 3 cm above the base (Diagram 2, C).
Point out the pattern for the pyramid: a slice through the apex is a triangle, and a slice that misses the apex usually has four or five sides. A pyramid has 5 faces, so its slices have at most 5 sides. A tilted cut that crosses the base and all four triangle faces makes a pentagon.
For the halfway slice, ask: "Why is it half as long, and not the same size?" The triangle faces lean in toward the apex, so the pyramid gets narrower as you go up. Halfway to the apex, it is half as wide in both directions. Compare this with the box, whose side faces go straight up.
Summary: shapes a slice can make
Shape of the slice
Box (right rectangular prism)
Right rectangular pyramid
Triangle
Yes: cut off a corner
Yes: slice straight down through the apex, or cut off a corner
Rectangle
Yes: slice parallel to any face
Yes: slice parallel to the base (smaller than the base)
Trapezoid
Yes: a tilted cut through 4 faces
Yes: slice straight down, missing the apex
Pentagon
Yes: a tilted cut through 5 faces
Yes: a tilted cut through all 5 faces
Hexagon
Yes: a tilted cut through all 6 faces
No: a pyramid has only 5 faces
Circle, or 7 or more sides
No
No
Guided Practice15 minutes
Pairs sketch each solid, draw the slice on the sketch, then name and measure the section. Reveal each answer after pairs have committed to one.
Guided practice with answers
Slice
Answer
A tissue box 24 cm long, 12 cm wide and 10 cm tall, sliced parallel to the base
A rectangle 24 cm by 12 cm
The same tissue box, sliced straight down parallel to the 12 cm by 10 cm end
A rectangle 12 cm by 10 cm
The same tissue box, cut from the top front edge down to the bottom back edge
A rectangle 24 cm long; its other side slants across the box, so it is longer than both 12 cm and 10 cm
A pyramid paperweight with a 5 cm by 5 cm base, 5 cm tall, sliced parallel to the base 1 cm below the apex
A square 1 cm by 1 cm: one fifth of the way down from the apex, each side is one fifth of 5 cm
The same paperweight, sliced straight down through the apex and two opposite corners of the base
A triangle 5 cm tall; its base is a diagonal of the square base (the segment joining opposite corners), which is longer than 5 cm
Listen for these errors: thinking every slice of a box is the same size as the base, naming a trapezoid a rectangle, and forgetting that the pyramid slices shrink in both directions, not just one.
Independent Practice15 minutes
Students sketch each solid, shade the slice, and describe it: shape, number of sides, parallel sides, and measurements where they can find them.
Independent practice with answers
Slice
Answer
A cube (a box whose edges all have the same length) with 4 cm edges, sliced parallel to a face
A square, 4 cm by 4 cm
The same cube, sliced through the midpoints (middle points) of 6 of its edges
A hexagon with six equal sides
A pyramid with a 10 cm by 10 cm base, 8 cm tall, sliced parallel to the base 6 cm up
A square 2.5 cm by 2.5 cm: 2 cm below the apex is one quarter of the height, so each side is one quarter of 10 cm
The same pyramid, sliced straight down through the apex, parallel to a base edge
A triangle with a 10 cm base and a height of 8 cm
The same pyramid, a tilted cut that crosses the base and all four triangle faces
A pentagon
Closure5 minutes
Exit ticket: (1) A tilted slice of a box crosses exactly 4 of its faces. How many sides does the slice have? (Answer: 4, one on each face it crosses.) (2) A pyramid has a 12 cm by 8 cm base. Describe the slice parallel to the base, halfway up. (Answer: a rectangle 6 cm by 4 cm.) (3) Sketch a slice of a box that is a triangle.
Differentiation Strategies
For Struggling Students
Let students hold and cut clay models (Activity 1) before they sketch, and trace each cut face on paper
Give a partly drawn box or pyramid, so students only draw the slice
Use a counting rule: count the faces the knife crosses; that is the number of sides of the slice
For Advanced Students
Ask for a slice of a box that is a square even though no face of the box is a square, and test it with clay
Ask how the area of a pyramid's slice parallel to the base changes from the base to halfway up (it becomes one quarter), and why
Challenge: sketch a tilted slice of a pyramid that is a pentagon, and mark where it crosses each of the 5 faces
Assessment Guidance
What to Look For
Check that students name the shape correctly and justify it by the faces the plane crosses, that they give measurements for slices parallel to a face or the base, and that they distinguish a trapezoid from a rectangle. For pyramids, look for slices that shrink in both directions and for the reason there is no six-sided slice.
02
Classroom Activities
3 Activities
1
Clay Slicing Lab
20 minPairs
Each pair shapes a clay box 6 cm long, 4 cm wide and 3 cm tall and a clay pyramid with a 6 cm by 4 cm base, 4 cm tall. They predict each slice, cut the clay with a piece of dental floss pulled tight, trace the cut face on paper, and measure it.
Procedure
For each cut below, sketch your prediction first, then cut, trace and measure
Box: (1) parallel to the base, (2) parallel to the 4 cm by 3 cm end, (3) cut off one corner, (4) a tilted cut from the top front edge to the bottom back edge
Pyramid: (5) parallel to the base, halfway up, (6) straight down through the apex, parallel to the 6 cm edge
Press the pieces back together (or reshape the clay) between cuts
Answer Key
(1) Rectangle 6 cm by 4 cm
(2) Rectangle 4 cm by 3 cm
(3) Triangle
(4) Rectangle 6 cm long; the other side is 5 cm, longer than both the 4 cm width and the 3 cm height
(5) Rectangle 3 cm by 2 cm
(6) Triangle with a 6 cm base and a height of 4 cm
Discussion Questions
Cuts (1), (2) and (4) all gave rectangles. Which of them had a side that was not an edge length of the box?
How many faces did the floss pass through in cut (3)? How does that match the number of sides?
Why was the pyramid slice in cut (5) smaller than the base, when the box slice in cut (1) was not?
Modification for Distance Learning
Students use a block of cheese, a stick of butter or a bar of soap at home, with an adult's help for any cutting, or they work only with sketches and the diagrams.
2
Can It Be a Slice? Card Sort
15 minGroups of 3
Each group gets 7 shape cards and a mat with four columns: "box and pyramid," "box only," "pyramid only" and "neither." Groups place each card, and for each one they either sketch a slice that makes it or explain why it is impossible.
Shape Cards
Triangle
Rectangle
Trapezoid
Pentagon
Hexagon
Circle
A shape with 7 sides
Answer Key
Box and pyramid: triangle, rectangle, trapezoid, pentagon
Box only: hexagon (a pyramid has only 5 faces)
Pyramid only: none of the cards
Neither: circle (every face is flat, so every side of a slice is straight) and the 7-sided shape (more sides than either solid has faces)
Discussion Questions
The "pyramid only" column stayed empty. Why can a box make every shape a pyramid can make?
For the trapezoid, where did you cut the box, and where did you cut the pyramid?
Challenge Variation
Add a "square" card. Groups decide whether a box that is 10 cm long, 6 cm wide and 4 cm tall, with no square face, can still have a square slice, and test the idea with clay.
3
Describe It, Draw It
15 minPairs
Partners sit back to back. One partner reads a description card aloud; the other sketches the solid and the slice and names the section. Then they switch. There are 6 cards.
Description Cards (answer key)
D1: A shoebox cut parallel to its base (a rectangle the size of the base)
D2: A pyramid cut parallel to its base, very close to the apex (a tiny rectangle with the same shape as the base)
D3: A box cut through the midpoints of 6 edges (a hexagon)
D4: A pyramid cut straight down through the apex (a triangle as tall as the pyramid)
D5: A pyramid cut straight down, missing the apex, parallel to a base edge (a trapezoid)
D6: A box with one corner cut off (a triangle)
Discussion Questions
Which words in the descriptions helped most: "parallel to the base," "through the apex," or the number of faces crossed?
D4 and D6 both give triangles. How are the two triangles different?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Four Slices of a Box
A block of cheese 10 cm long, 6 cm wide and 4 cm tall, sliced four ways. Slices parallel to a face (A, B) are rectangles the size of that face. Tilted slices can be triangles (C, crossing 3 faces) or hexagons (D, crossing all 6 faces, here through the midpoints of 6 edges). Dashed lines are hidden edges. Drawn to scale in a slanted 3D view.
Diagram 2: Three Slices of a Pyramid
A right rectangular pyramid with an 8 cm by 6 cm base, 6 cm tall. A: parallel to the base, halfway up, the slice is a 4 cm by 3 cm rectangle. B: straight down through the apex, parallel to the 8 cm edge, the slice is a triangle 8 cm wide and 6 cm tall. C: straight down, 1.5 cm in front of the center, the slice is a trapezoid. The true shape of each slice is drawn below it, to scale. In the slanted view, widths and heights are to scale and depth is drawn at half scale.
04
Homework Assignment
~30 min
7.G.A.3 Homework: Describing Slices of Prisms and Pyramids
Directions: For each problem, sketch the solid and shade the slice. Describe every slice by its shape, its number of sides and its measurements when you can find them, and explain your reasoning.
Part 1: Slices of Boxes (Problems 1-3)
A shoebox is 30 cm long, 18 cm wide and 12 cm tall. Describe the slice, with its dimensions: (a) parallel to the base (b) straight down, parallel to the 18 cm by 12 cm end (c) straight down, parallel to the 30 cm by 12 cm side. Then find the area of slice (a).
Sketch a box three times. On the sketches, draw a slice that is (a) a triangle (b) a pentagon (c) a hexagon. For each one, say how many faces of the box the slice crosses.
True or false? Give a reason for each. (a) Every slice of a box that is parallel to a face is a rectangle. (b) A slice of a box can have 7 sides. (c) A slice of a box can be a triangle.
Part 2: Slices of Pyramids (Problems 4-6)
A right rectangular pyramid has a 16 cm by 10 cm base and is 20 cm tall. Describe the slices parallel to the base at 5 cm, 10 cm and 15 cm above the base. What pattern do you see?
A right rectangular pyramid has a 14 cm by 6 cm base and is 9 cm tall. Describe the two slices that go straight down through the apex: one parallel to the 14 cm edge and one parallel to the 6 cm edge. Find the area of each.
Make a list of the shapes a slice of a box can be, and a list for a slice of a right rectangular pyramid. Which shape is on one list but not the other? Explain why.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sketches
Solid and slice drawn clearly, slice in the right place
Slice drawn but hard to read
No sketch
Shape Names
Every section named correctly, with its number of sides
One or two names wrong
Most names wrong
Measurements
Dimensions and areas correct where they can be found
One or two errors
Measurements missing
Reasoning
Explains by faces crossed, parallel faces or shrinking toward the apex
Some reasons missing
No reasons given
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Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. A quick sketch helps for most of them. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A wooden block is a box 12 cm long, 5 cm wide and 3 cm tall, resting on its 12 cm by 5 cm face. What is the slice parallel to the base?
Answer: B
A slice parallel to the base is a copy of the base: 12 cm by 5 cm. Choice A is the slice parallel to the long side face. Choice C is the slice parallel to the end. Choice D halves both base lengths, as if the block shrank toward the top like a pyramid.
Question 2 of 20 · Multiple Choice
The same 12 cm by 5 cm by 3 cm block is sliced straight down, parallel to its smallest faces. What is the slice?
Answer: D
The smallest faces are the 5 cm by 3 cm ends, so a slice parallel to them is a 5 cm by 3 cm rectangle. Choice A is the base. Choice B is parallel to the long side faces. Choice C uses half the length, 6 cm, but the slice goes across the block, not along it.
Question 3 of 20 · Multiple Choice
Which shape can NOT be a plane section of a right rectangular pyramid?
Answer: C
Each side of a section lies on a different face, and a pyramid has only 5 faces (1 base and 4 triangles), so no section has 6 sides. Choice A comes from a slice through the apex. Choice B comes from a slice straight down that misses the apex. Choice D comes from a tilted cut through all 5 faces; a student who counts only the 4 triangle faces misses it.
Question 4 of 20 · Multiple Choice
Which shape can NOT be a plane section of a box (a right rectangular prism)?
Answer: A
A box has 6 faces, and each side of a section lies on a different face, so a section has at most 6 sides. Choice B is possible: a tilted cut through all 6 faces. Choice C is possible: a tilted cut through 5 faces. Choice D is possible: cut off one corner. Thinking only of cuts parallel to a face leads to rejecting B, C and D.
Question 5 of 20 · Multiple Choice
A right rectangular pyramid has a 10 cm by 8 cm base and is 12 cm tall. What is the slice parallel to the base, halfway up?
Answer: C
Halfway to the apex, the pyramid is half as wide in both directions: 10 ÷ 2 = 5 cm and 8 ÷ 2 = 4 cm. Choice A treats the pyramid like a box. Choice B halves only one direction. Choice D divides by 4, as if the slice were three quarters of the way up.
Question 6 of 20 · Multiple Choice
The same pyramid (10 cm by 8 cm base, 12 cm tall) is sliced straight down through the apex, parallel to the 10 cm edge. What is the slice?
Answer: A
A slice through the apex is a triangle. Parallel to the 10 cm edge, it crosses the base along a 10 cm line, and its top point is the apex, 12 cm up. Choice B is the slice parallel to the 8 cm edge. Choice C forgets that the faces meet at the apex. Choice D describes a slice that misses the apex.
Question 7 of 20 · Multiple Choice
A right rectangular pyramid is sliced straight down, parallel to one base edge, but the slice does not pass through the apex. What shape is the slice?
Answer: D
The slice has a long bottom side on the base and a shorter top side where it leaves the front and back faces, and these two sides are parallel. The other two sides slant inward, so it is a trapezoid. Choice A would need the slice to go through the apex. Choice B ignores that the faces lean in. Choice C needs a tilted cut through 5 faces.
Question 8 of 20 · Multiple Choice
A plane cuts off one corner of a box, crossing the three faces that meet at that corner. What is the section?
Answer: B
The plane crosses 3 faces, and each face holds one side of the section, so the section has 3 sides: a triangle. Choice A is what a straight cut parallel to a face gives. Choices C and D need the plane to cross 5 or 6 faces.
Question 9 of 20 · Multiple Choice
A tilted plane crosses all six faces of a box. What is the section?
Answer: C
Six faces crossed means six sides: a hexagon. Choice A has only 4 sides, from crossing 4 faces. Choice B comes from crossing 5 faces. Choice D, an 8-sided shape, would need 8 faces, but a box has only 6.
Question 10 of 20 · Multiple Choice
A slice parallel to the base of a right rectangular pyramid moves slowly from the base up toward the apex. What happens to the slice?
Answer: B
The four triangle faces lean in toward the apex, so both side lengths shrink by the same fraction: the slice is always a smaller copy of the base, until it becomes a point at the apex. Choice A describes a box. Choice C confuses it with a slice through the apex. Choice D shrinks only one direction.
Question 11 of 20 · Multiple Choice
A sheet cake is a box 33 cm long, 23 cm wide and 5 cm tall. Ana cuts straight down from one corner of the top to the opposite corner. What is the cut face?
Answer: D
A straight-down cut through a box is a rectangle as tall as the box, 5 cm. Its length is the diagonal of the top, corner to corner, which is longer than the 33 cm side. Choice A comes from looking only at the top, where the cut splits the rectangle into two triangles. Choices B and C are cuts parallel to a side, not corner to corner.
Question 12 of 20 · Multiple Choice
A stick of butter is a box about 12 cm long, 3 cm wide and 3 cm tall. You cut it straight across, parallel to its small end. What is the cut face?
Answer: A
A slice parallel to the end is a copy of the end, which is 3 cm by 3 cm: a square. A square is a special rectangle. Choice B is a slice along the length of the stick. Choice C would need a cut that crosses only 3 faces. Choice D is impossible, because every face of a box is flat, so every side of a slice is straight.
Question 13 of 20 · Multiple Choice
A right rectangular pyramid has an 8 cm by 4 cm base and is 10 cm tall. What is the area of its slice parallel to the base, halfway up?
Answer: C
Halfway up, each side is half as long: 4 cm by 2 cm, so the area is 4 × 2 = 8 cm², one quarter of the 32 cm² base. Choice A is the base area. Choice B halves the area instead of halving each side. Choice D uses the triangle area formula, ½ × 4 × 2 = 4, but the slice is a rectangle.
Question 14 of 20 · Multiple Choice
Which slice of a right rectangular pyramid is a triangle exactly as tall as the pyramid?
Answer: B
A straight-down slice through the apex runs from the base up to the apex, so its height is the height of the pyramid. Choice A is a tiny rectangle. Choice C misses the apex, so it is a trapezoid. Choice D is a triangle, but a small one near the corner, not as tall as the pyramid.
Question 15 of 20 · Short Answer
A brick is a box 20 cm long, 9 cm wide and 6 cm tall. Describe the three different slices you can make parallel to its faces, and find the area of each.
Parallel to the base: a rectangle 20 cm by 9 cm, area 180 cm². Parallel to the long side face: a rectangle 20 cm by 6 cm, area 120 cm². Parallel to the end: a rectangle 9 cm by 6 cm, area 54 cm². Each slice is a copy of the face it is parallel to.
Question 16 of 20 · Short Answer
A right rectangular pyramid has a 21 cm by 9 cm base and is 12 cm tall. Describe the slice parallel to the base, 8 cm above the base.
8 cm above the base is 4 cm below the apex, which is 4/12 = 1/3 of the way down from the apex. Each side is 1/3 of the base side: a rectangle 7 cm by 3 cm, the same shape as the base.
Question 17 of 20 · Short Answer
A right rectangular pyramid is sliced by a plane. What is the greatest number of sides the slice can have? Explain.
Each side of a section is a straight segment where the plane crosses one face, and the plane can cross each flat face only once. A right rectangular pyramid has only 5 faces (the base and 4 triangles), so a section has at most 5 sides. A tilted cut that crosses the base and all four triangle faces gives a pentagon, so 5 sides really happens.
Question 18 of 20 · Short Answer
Describe how to slice a box so that the section is a pentagon.
Use a tilted cut that crosses exactly 5 of the 6 faces. For example, start near one top corner and cut down at a slant so that the plane crosses the top, two side faces next to that corner, the bottom, and one more side face, but misses the face opposite that corner. Each face crossed gives one side, so the section has 5 sides.
Question 19 of 20 · Short Answer
A right rectangular pyramid has a 12 cm by 5 cm base and is 9 cm tall. It is sliced straight down through the apex, parallel to the 5 cm edge. Describe the slice and find its area.
The slice is a triangle with a 5 cm base and a height of 9 cm, the height of the pyramid. Its area is ½ × 5 × 9 = 22.5 cm².
Question 20 of 20 · Short Answer
A spaghetti box is 26 cm long, 7 cm wide and 3 cm tall and lies flat on a table. Kai says that slices parallel to the table get smaller near the top, like the slices of a pyramid. Is he right? Explain.
No. The side faces of a box stand straight up, so every slice parallel to the table is the same 26 cm by 7 cm rectangle, at any height. Pyramid slices shrink because the triangle faces lean in toward the apex.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.G.A.3 mean?
7.G.A.3 means students can describe the flat shape made when a plane slices a solid, especially a box or a right rectangular pyramid. For example, a slice of a box parallel to its base is a rectangle the size of the base, and a slice of a pyramid straight down through its apex is a triangle.
What grade is 7.G.A.3, and what comes after it?
It is a grade 7 standard in the Geometry domain. In grade 8, students study figures that are scaled copies of each other (8.G.A.4), which explains why the slices of a pyramid parallel to its base all have the same shape. In high school geometry, students identify cross sections of other solids, such as cylinders and cones (HSG.GMD.B.4).
What is a plane section?
A plane section is the flat shape where a plane cuts through a solid. It is the same as a cross section. Think of the face you see after cutting straight through a block of cheese: its outline is the plane section.
What is the difference between a right rectangular prism and a right rectangular pyramid?
A right rectangular prism is a box: 6 rectangle faces, with the side faces standing straight up from the base. A right rectangular pyramid has one rectangle base and 4 triangle faces that meet at the apex, directly above the center of the base. "Right" means the solid is not leaning to one side.
Why can a slice of a box have at most 6 sides?
Each side of the slice is where the plane crosses one face, and a flat plane crosses a flat face in at most one segment. A box has 6 faces, so the slice has at most 6 sides. The same reasoning limits a pyramid, which has 5 faces, to at most 5 sides.
Why do pyramid slices parallel to the base get smaller?
The triangle faces lean inward and meet at the apex, so the pyramid gets narrower as you go up. A slice halfway up is half as long and half as wide as the base, and a slice near the apex is tiny. The slices keep the shape of the base because both directions shrink by the same fraction.
Does 7.G.A.3 include cylinders, cones or spheres?
No. The standard names right rectangular prisms and right rectangular pyramids. Slices of cylinders, cones and spheres, which can give circles and ovals, are part of high school geometry (HSG.GMD.B.4). Teachers can mention them as a preview, but grade 7 practice should stay with boxes and pyramids.
How can students see slices they cannot cut?
Clay models cut with dental floss show real slices, and a clear box half filled with colored water shows a slice at the water line when you tilt it. Sketching also helps: draw the solid, mark where the plane crosses each edge, and join the marks with straight lines.
What mistakes should teachers watch for?
A common mistake is calling every four-sided slice a rectangle, including trapezoids from pyramids. Other frequent errors are thinking box slices shrink near the top, forgetting that pyramid slices shrink in both directions, and claiming a hexagon can come from a pyramid.
How can parents help with 7.G.A.3 at home?
When you slice cheese, bread, tofu or a stick of butter, ask your child to predict the shape of the cut face first. Try a straight cut and a slanted one, and talk about why the shapes differ. Cutting off one corner of a block of cheese shows a triangle.
07
Related Standards
6 standards
These standards connect to 7.G.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.G.A.4Prerequisite
Represent 3D figures with nets of rectangles and triangles and find surface area
