Cross Sections of Solids
Learn how slicing a right rectangular prism or pyramid with a plane parallel or perpendicular to the base creates 2D cross sections — rectangles, squares, and trapezoids.
What you'll do in this lesson
A voice-first session with the Crimsora tutor on Cross Sections of Solids, then targeted practice and FRQs — with the tutor adapting to where you get stuck.
What this lesson covers
Imagine a block of cheese and a very sharp knife. Wherever you cut, the flat face you expose is a two-dimensional shape called a cross section. Change the direction of the cut, and the shape changes too. Slice a rectangular prism straight across and you see a rectangle; slice a pyramid partway up and you see a smaller version of its base.
In this lesson you will learn to picture and name the cross sections formed when a plane slices a right rectangular prism or a right rectangular pyramid, either parallel to the base or perpendicular to it. You will practice describing not just the shape's name but its actual dimensions, and you will see why the pyramid behaves so differently from the prism. This kind of spatial reasoning shows up later in surface area and volume work, and in real jobs like architecture, medical imaging, and manufacturing.
In this lesson you will learn to picture and name the cross sections formed when a plane slices a right rectangular prism or a right rectangular pyramid, either parallel to the base or perpendicular to it. You will practice describing not just the shape's name but its actual dimensions, and you will see why the pyramid behaves so differently from the prism. This kind of spatial reasoning shows up later in surface area and volume work, and in real jobs like architecture, medical imaging, and manufacturing.
What a Cross Section Actually Is
A cross section is the two-dimensional figure formed where a plane intersects a three-dimensional solid. Think of the plane as an infinitely thin, infinitely wide flat sheet passing through the solid. Everywhere the sheet touches the inside of the solid, you get a flat region — that region is the cross section.
Two images help. The first is slicing: cut the solid along the plane, pull the top piece away, and look straight down at the newly exposed flat face. The second is dipping: lower the solid into water and look at the shape of the waterline on the solid's surface at one instant. Both describe the same figure.
Students often confuse a cross section with a face, a shadow, or a net. A face is part of the solid's outside surface and already exists before you cut. A shadow is a projection onto a wall and can be distorted. A net is the flattened-out surface of the whole solid. A cross section is different from all three: it is a brand-new flat shape that appears only where the cutting plane meets the interior.
The orientation of the plane is what controls the answer. In this lesson the plane is always in one of two special positions relative to the base: parallel to the base (like a shelf, never touching the base) or perpendicular to the base (standing straight up, at a right angle to it). Slanted planes make other shapes, but the two special orientations are predictable enough to reason about exactly.
Two images help. The first is slicing: cut the solid along the plane, pull the top piece away, and look straight down at the newly exposed flat face. The second is dipping: lower the solid into water and look at the shape of the waterline on the solid's surface at one instant. Both describe the same figure.
Students often confuse a cross section with a face, a shadow, or a net. A face is part of the solid's outside surface and already exists before you cut. A shadow is a projection onto a wall and can be distorted. A net is the flattened-out surface of the whole solid. A cross section is different from all three: it is a brand-new flat shape that appears only where the cutting plane meets the interior.
The orientation of the plane is what controls the answer. In this lesson the plane is always in one of two special positions relative to the base: parallel to the base (like a shelf, never touching the base) or perpendicular to the base (standing straight up, at a right angle to it). Slanted planes make other shapes, but the two special orientations are predictable enough to reason about exactly.
Slicing a Right Rectangular Prism
A right rectangular prism is a box: six rectangular faces, with the lateral edges perpendicular to the base. Suppose a box is 8 cm long, 5 cm wide, and 3 cm tall, with the 8 by 5 rectangle as the base.
A cut parallel to the base is a horizontal slice. No matter how high you cut, the cross section is a rectangle that is exactly congruent to the base: 8 cm by 5 cm. That is because the vertical walls of the box do not lean in or out, so the horizontal outline never changes as you move up. This is the defining feature of a prism.
A cut perpendicular to the base stands upright. It is still a rectangle, but its dimensions depend on which direction the plane faces. A vertical cut running the long way gives an 8 cm by 3 cm rectangle; a vertical cut running the short way gives a 5 cm by 3 cm rectangle. One dimension is always the height of the prism, and the other is a horizontal dimension of the base.
A common wrong answer is claiming a parallel cut higher up gives a smaller rectangle. It does not. Nothing shrinks in a prism, only in a pyramid.
A cut parallel to the base is a horizontal slice. No matter how high you cut, the cross section is a rectangle that is exactly congruent to the base: 8 cm by 5 cm. That is because the vertical walls of the box do not lean in or out, so the horizontal outline never changes as you move up. This is the defining feature of a prism.
A cut perpendicular to the base stands upright. It is still a rectangle, but its dimensions depend on which direction the plane faces. A vertical cut running the long way gives an 8 cm by 3 cm rectangle; a vertical cut running the short way gives a 5 cm by 3 cm rectangle. One dimension is always the height of the prism, and the other is a horizontal dimension of the base.
| Plane position | Shape | Dimensions |
|---|---|---|
| Parallel to base | Rectangle | 8 cm by 5 cm (congruent to base) |
| Perpendicular, parallel to the 8 cm edge | Rectangle | 8 cm by 3 cm |
| Perpendicular, parallel to the 5 cm edge | Rectangle | 5 cm by 3 cm |
Slicing a Right Rectangular Pyramid
A right rectangular pyramid has a rectangular base and an apex directly above the center of that base. The four lateral faces are triangles that lean inward toward the apex, and this leaning is what makes the pyramid's cross sections behave differently from a prism's.
A cut parallel to the base produces a rectangle with the same shape as the base but smaller — the two rectangles are similar, not congruent. The higher you slice, the smaller the rectangle, until at the apex the cross section shrinks to a single point. If the base is a square, every parallel cross section is a smaller square.
A cut perpendicular to the base gives two possible answers, and this is where students most often go wrong. If the vertical plane passes through the apex, the cross section is a triangle: its bottom edge lies in the base and its top vertex is the apex. If the vertical plane misses the apex, the slanted lateral faces cut off the top, so the cross section is a trapezoid — a long bottom side in the base, a shorter top side, and two slanted legs.
Before answering any pyramid question, ask yourself one thing: does the plane hit the apex? That single question decides between triangle and trapezoid.
A cut parallel to the base produces a rectangle with the same shape as the base but smaller — the two rectangles are similar, not congruent. The higher you slice, the smaller the rectangle, until at the apex the cross section shrinks to a single point. If the base is a square, every parallel cross section is a smaller square.
A cut perpendicular to the base gives two possible answers, and this is where students most often go wrong. If the vertical plane passes through the apex, the cross section is a triangle: its bottom edge lies in the base and its top vertex is the apex. If the vertical plane misses the apex, the slanted lateral faces cut off the top, so the cross section is a trapezoid — a long bottom side in the base, a shorter top side, and two slanted legs.
| Plane position | Cross section |
|---|---|
| Parallel to base, below apex | Rectangle similar to base, smaller |
| Perpendicular, through apex | Triangle |
| Perpendicular, not through apex | Trapezoid |
Strategies for Picturing the Slice
Spatial questions get much easier when you stop trying to hold the whole picture in your head and instead build it in steps.
Start by drawing the solid and marking the base clearly. Then draw the cutting plane as a straight line across your sketch — a horizontal line for a parallel cut, a vertical line for a perpendicular cut. Next, trace only the points where that line crosses the solid's edges. Those crossing points are the vertices of your cross section. Counting them tells you a lot: four crossing points on a prism means a four-sided figure, three on a pyramid through the apex means a triangle.
Physical models are worth the effort. A stack of index cards behaves like a prism: pull the stack apart anywhere and every card is identical, which shows why parallel cross sections of a prism are congruent. A ball of clay shaped into a pyramid can be cut with dental floss so you can look at the exposed face directly.
Finally, describe the result completely. "A rectangle" is a partial answer; "a rectangle 6 cm by 4 cm, congruent to the base" is a complete one. When a question asks about a pyramid, name the shape and say whether the plane passed through the apex, since that is the reasoning that justifies your answer. Students who write down the plane's position first almost never mix up trapezoid and triangle.
Start by drawing the solid and marking the base clearly. Then draw the cutting plane as a straight line across your sketch — a horizontal line for a parallel cut, a vertical line for a perpendicular cut. Next, trace only the points where that line crosses the solid's edges. Those crossing points are the vertices of your cross section. Counting them tells you a lot: four crossing points on a prism means a four-sided figure, three on a pyramid through the apex means a triangle.
Physical models are worth the effort. A stack of index cards behaves like a prism: pull the stack apart anywhere and every card is identical, which shows why parallel cross sections of a prism are congruent. A ball of clay shaped into a pyramid can be cut with dental floss so you can look at the exposed face directly.
Finally, describe the result completely. "A rectangle" is a partial answer; "a rectangle 6 cm by 4 cm, congruent to the base" is a complete one. When a question asks about a pyramid, name the shape and say whether the plane passed through the apex, since that is the reasoning that justifies your answer. Students who write down the plane's position first almost never mix up trapezoid and triangle.
Key terms
- Cross section.
- The two-dimensional figure formed where a plane intersects a three-dimensional solid.
- Plane.
- A flat surface that extends forever in all directions and has no thickness; used as the cutting tool in slicing problems.
- Right rectangular prism.
- A solid with two congruent parallel rectangular bases joined by lateral edges perpendicular to those bases; a box.
- Right rectangular pyramid.
- A solid with a rectangular base and an apex located directly above the center of the base, joined by four triangular faces.
- Apex.
- The single point of a pyramid where all the triangular lateral faces meet.
- Parallel to the base.
- Describes a cutting plane that stays the same distance from the base everywhere, so it never meets the base.
- Perpendicular to the base.
- Describes a cutting plane that meets the base at a right angle, standing upright through the solid.
- Trapezoid.
- A four-sided figure with exactly one pair of parallel sides; the cross section formed by a vertical cut through a pyramid that misses the apex.
Worked example
A right rectangular pyramid has a base 10 in. long and 6 in. wide, with a height of 12 in. Describe the cross section formed by (a) a plane parallel to the base halfway up the pyramid, and (b) a vertical plane that passes 2 in. away from the apex, parallel to the 6 in. edge of the base.
Part (a): The plane is parallel to the base, so the cross section is a rectangle similar to the base. At halfway up, every horizontal dimension has shrunk by the same factor as the distance remaining to the apex. Half the height means the scale factor is , so the length is in. and the width is in. The cross section is a 5 in. by 3 in. rectangle, similar to the base with scale factor .
Part (b): First ask the key question — does the plane pass through the apex? No, it passes 2 in. away from it. So the slanted lateral faces cut off the top of the slice, leaving a shorter top edge. The cross section is a trapezoid. Its bottom side lies in the base, its top side is shorter, and its two legs lie along the slanted triangular faces.
Notice how each answer names the figure and explains why. In part (a) the justification is that the plane is parallel to the base; in part (b) it is that the plane misses the apex. If the plane in part (b) had gone through the apex instead, the two slanted legs would have met at a single point and the answer would have been a triangle.
Part (b): First ask the key question — does the plane pass through the apex? No, it passes 2 in. away from it. So the slanted lateral faces cut off the top of the slice, leaving a shorter top edge. The cross section is a trapezoid. Its bottom side lies in the base, its top side is shorter, and its two legs lie along the slanted triangular faces.
Notice how each answer names the figure and explains why. In part (a) the justification is that the plane is parallel to the base; in part (b) it is that the plane misses the apex. If the plane in part (b) had gone through the apex instead, the two slanted legs would have met at a single point and the answer would have been a triangle.
Practice questions
A right rectangular prism has a base 9 cm by 4 cm and a height of 7 cm. A plane slices the prism parallel to the base, 5 cm above the base. What is the cross section?
- A rectangle 9 cm by 4 cm
- A rectangle 9 cm by 7 cm
- A rectangle 4 cm by 5 cm
- A rectangle 9 cm by 2 cm
Answer: A rectangle 9 cm by 4 cm
In a right rectangular prism the vertical walls do not lean, so a horizontal slice always produces a rectangle congruent to the base — 9 cm by 4 cm — no matter what height you cut at. The 5 cm height is extra information that does not change the shape. The choice with a 7 cm side would come from a perpendicular cut instead, and 9 cm by 2 cm mistakenly subtracts the slice height from a base dimension, which has no geometric meaning here.
Mia says that any vertical slice through a right rectangular pyramid gives a triangle, because the pyramid's side faces are triangles. Explain what is correct and what is incorrect about her reasoning, and describe when each cross section actually occurs.
Answer: Mia is only sometimes right. A vertical (perpendicular to the base) slice through a right rectangular pyramid gives a triangle only when the cutting plane passes through the apex. If the plane misses the apex, the slanted lateral faces cut off the top of the slice, so the cross section has a long bottom side in the base, a shorter parallel top side, and two slanted legs — a trapezoid.
Mia's correct instinct is that the pyramid narrows as you go up, so vertical cross sections are narrower at the top. Her error is assuming they always narrow all the way to a point. Only a plane containing the apex reaches that single point. Any other vertical plane stops short: at the top of that slice the pyramid still has some width, giving a second parallel side and therefore a trapezoid. The test to apply every time is simply whether the plane contains the apex.
A right rectangular pyramid has a square base with sides 8 m and a height of 8 m. A plane parallel to the base cuts the pyramid 6 m above the base. Find the side length of the cross section and name the figure.
Answer: A square with sides 2 m.
Because the plane is parallel to the base, the cross section is similar to the base, so a square base gives a square cross section. The slice is 6 m up on an 8 m tall pyramid, leaving m to the apex, so the scale factor is . Multiplying: m. A frequent error is using as the scale factor; remember the cross section shrinks toward the apex, so compare the remaining height above the slice to the full height.
FAQ
- What is the difference between a cross section and a face of a solid?
- A face is part of the outside surface of the solid and exists before you do anything to it. A cross section is a new flat shape revealed only where a cutting plane passes through the interior. Sometimes they look alike — a parallel slice of a prism matches its base face exactly — but they are formed in completely different ways.
- Why do parallel cross sections of a prism stay the same size but shrink in a pyramid?
- In a right rectangular prism the lateral edges are perpendicular to the base, so the walls go straight up and the horizontal outline never changes. In a pyramid the lateral edges lean inward toward the apex, so the horizontal outline gets smaller the higher you cut, eventually shrinking to a point at the apex.
- Can slicing a rectangular prism ever give a triangle?
- Not with a plane parallel or perpendicular to the base — those always produce rectangles. A slanted plane that cuts across one corner of the box can produce a triangle, but slanted cuts are beyond what this lesson asks you to describe.
- How do I decide between triangle and trapezoid for a pyramid?
- Ask one question: does the cutting plane pass through the apex? If yes, the two slanted sides meet at that single point and you get a triangle. If no, the top of the slice still has width, giving a shorter parallel top side, so you get a trapezoid.
Learn this with a teacher, not a page
The Crimsora tutor teaches Cross Sections of Solids live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.