Cross Sections of 3D Shapes: Examples

A cross section is the flat shape you get when a plane slices through a solid. Slicing a cylinder parallel to its base gives a circle, and slicing a cube parallel to a face gives a square. Cross sections also work in reverse: spinning a flat shape around a line sweeps out a solid.

Before this lesson: You should know Points, Lines and Planes, Circumference and Area of Circles and Similar Polygons.

What Is a Cross Section?

Think of slicing a loaf of bread. Each slice shows a flat face. In geometry, the knife is a plane, a flat surface that goes on forever. The shape where the plane meets the solid is the cross section.

The shape depends on two things: the solid, and the direction of the cut. A cut parallel to the base usually copies the base. A perpendicular cut, straight up and down, often gives a rectangle or a triangle. A slanted cut can give other shapes.

Three solids, each cut by a flat plane parallel to its base, with the cut shaded orange. The cube gives a square cross section. The cylinder gives a circle the same size as its base. The cone gives a smaller circle. A second row shows other cuts: a slanted cut across a corner of the cube gives a triangle, a vertical cut through the cylinder axis gives a rectangle, and a vertical cut through the cone tip gives a triangle. — The same solid can have different cross sections, depending on the direction of the cut.

Common Cross Sections

Solid	Cut parallel to the base	Cut perpendicular to the base	Slanted cut
Cube or rectangular prism	Rectangle (square for a cube)	Rectangle	Triangle (at a corner), rectangle, or a shape with up to 6 sides
Cylinder	Circle, same size as the base	Rectangle	Ellipse (an oval)
Cone	Circle, smaller near the tip	Triangle (through the tip)	Ellipse or a curved shape
Square pyramid	Square, smaller near the tip	Triangle (through the tip)	Quadrilateral or triangle
Sphere	Circle (every cut)	Circle	Circle

A cut through a sphere is always a circle. The biggest circle comes from a cut through the center. It is called a great circle.

Solids of Rotation

Spin a flat shape around a line, called the axis, and it sweeps out a solid. This is a solid of revolution.

Left: a rectangle 3 units wide and 5 units tall spins around its 5-unit side and sweeps out a cylinder with radius 3 and height 5. Right: a right triangle with legs 4 and 6 spins around its 6-unit leg and sweeps out a cone with radius 4 and height 6. — Spinning a rectangle makes a cylinder. Spinning a right triangle around a leg makes a cone.

A rectangle spun around one side makes a cylinder. The side on the axis is the height, and the other side is the radius.
A right triangle spun around a leg makes a cone. The leg on the axis is the height, and the other leg is the radius.
A semicircle spun around its diameter makes a sphere.

How to Find a Cross Section

Picture the solid and the direction of the plane: parallel, perpendicular or slanted.
Name the shape of the cut, using the table above.
Find its dimensions. A cut parallel to a cylinder’s base matches the base. For a cone or pyramid, use similar triangles.
If asked, find the area of that shape.

Worked Examples

Example 1: Name the cross sections of a cube

Name the cross section of a cube for each cut: (a) parallel to a face, (b) perpendicular to the top through two opposite edges, (c) a slanted cut that cuts off one corner.

Part (a). A cut parallel to a face makes a copy of that face: a square.
Part (b). The cut goes straight down through two opposite edges, so it has two edges of the cube and two face diagonals as sides: a rectangle.
Part (c). Cutting off one corner meets three faces near that corner, so the cut has three sides: a triangle.

Answer: (a) square (b) rectangle (c) triangle

Example 2: Area of cylinder cross sections

A cylinder has radius 4 cm and height 10 cm. Find the area of (a) a cut parallel to the base and (b) a vertical cut through the center (the axis).

Part (a). The cut is a circle with the same radius as the base: $A = \pi(4)^2$
Square: $A = 16\pi$
Use a calculator: $A \approx 50.27$ cm²
Part (b). The cut is a rectangle. Its width is the diameter: $2 \cdot 4 = 8$
Its height is the cylinder’s height, 10. Multiply: $A = 8 \cdot 10 = 80$ cm²

Answer: (a) $16\pi \approx 50.27$ cm² (b) 80 cm²

Example 3: Name a solid of revolution

A right triangle has legs 4 and 6. It spins around the leg of length 6. Describe the solid.

The leg on the axis becomes the height: $h = 6$
The other leg sweeps out the base, so it is the radius: $r = 4$
The hypotenuse sweeps out the slanted side.

Answer: A cone with radius 4 and height 6.

Example 4: Cross section of a sphere

A sphere has radius 10 in. A plane cuts it 6 in from the center. Find the area of the cross section.

The cut is a circle. Its radius $x$ , the 6 in distance and the sphere’s radius 10 form a right triangle with hypotenuse 10: $x^2 + 6^2 = 10^2$
Square: $x^2 + 36 = 100$
Subtract 36 from both sides: $x^2 = 64$
Take the positive square root (see the Pythagorean theorem): $x = 8$
Find the area of the circle: $A = \pi(8)^2 = 64\pi$
Use a calculator: $A \approx 201.06$

Answer: $64\pi \approx 201.06$ in²

Example 5: Word problem with a traffic cone

A traffic cone is 60 cm tall and has a base radius of 15 cm. It is cut parallel to the base, 20 cm above the base. What is the area of the cross section?

The cut is 20 cm above the base, so it is $60 - 20 = 40$ cm below the tip.
The small cone above the cut is similar to the whole cone (see similar polygons). Set up a proportion: $\frac{r}{15} = \frac{40}{60}$
Multiply both sides by 15: $r = 15 \cdot \frac{40}{60}$
Simplify: $r = 10$
Find the area of the circle: $A = \pi(10)^2 = 100\pi$
Use a calculator: $A \approx 314.16$

Answer: $100\pi \approx 314.16$ cm²

Common Mistakes

Mistake	Example of the error	Fix
Thinking every cut of a cylinder is a circle	Saying a vertical cut through a can is a circle	Only cuts parallel to the base are circles. A vertical cut is a rectangle.
Using the radius as the width of the rectangle	Width 4 for a cylinder with radius 4	A cut through the axis is as wide as the diameter, 8.
Using the full base for every cone cut	Radius 15 at any height	Cone cuts shrink toward the tip. Use similar triangles.
Mixing up the height and radius when spinning	Spinning around the 6-unit leg and calling 6 the radius	The side on the axis is the height.
Measuring from the wrong end	Using 20 cm from the tip instead of the base	Find the distance from the tip: $60 - 20 = 40$ .

Practice Problems

A square pyramid is cut parallel to its base. What shape is the cross section?
A square pyramid is cut by a vertical plane through its tip and the midpoints of two opposite base edges. What shape is the cross section?
What shape is every cross section of a sphere? Where is the largest one?
A cylinder has radius 5 m and height 12 m. Find the area of a cross section parallel to the base.
For the same cylinder, find the area of a vertical cross section through the axis.
A rectangle that is 2 units by 7 units spins around its 7-unit side. Describe the solid.
A right triangle with legs 5 and 12 spins around the leg of length 12. Describe the solid. What if it spins around the leg of length 5?
A semicircle spins around its diameter. What solid does it make?
A sphere has radius 13. A plane cuts it 5 units from the center. Find the radius and the area of the cross section.
A cone is 9 cm tall with base radius 6 cm. It is cut parallel to the base, 3 cm below the tip. Find the area of the cross section.

Show answers

A cut parallel to the base copies the base, so it is a smaller square.
The cut passes through the tip and splits the base along a middle line, so it is a triangle.
Every cross section of a sphere is a circle. The largest, a great circle, comes from a cut through the center.
The cut is a circle with radius 5: $A = \pi(5)^2$
Square: $A = 25\pi$
Use a calculator: $A \approx 78.54$ m²
The cut is a rectangle. Its width is the diameter: $2 \cdot 5 = 10$
Multiply by the height: $A = 10 \cdot 12 = 120$ m²
The 7-unit side is on the axis, so it is the height. The 2-unit side is the radius. The solid is a cylinder with radius 2 and height 7.
Around the leg of 12: a cone with height 12 and radius 5.
Around the leg of 5: a cone with height 5 and radius 12.
A sphere.
Write the Pythagorean theorem: $x^2 + 5^2 = 13^2$
Square: $x^2 + 25 = 169$
Subtract 25 from both sides: $x^2 = 144$
Take the positive square root: $x = 12$
Find the area: $A = \pi(12)^2 = 144\pi$
Use a calculator: $A \approx 452.39$
The small cone above the cut is similar to the whole cone: $\frac{r}{6} = \frac{3}{9}$
Multiply both sides by 6: $r = 6 \cdot \frac{3}{9}$
Simplify: $r = 2$
Find the area: $A = \pi(2)^2 = 4\pi$
Use a calculator: $A \approx 12.57$ cm²

FAQ

What is a cross section in geometry?

It is the two-dimensional shape you see where a plane cuts through a three-dimensional solid.

What are the cross sections of a cylinder?

A cut parallel to the base is a circle. A vertical cut is a rectangle. A slanted cut that doesn’t touch the bases is an ellipse.

What solid do you get when you rotate a rectangle?

Spinning a rectangle around one of its sides makes a cylinder. The side on the axis is the height and the other side is the radius.

Can a cross section of a cube be a hexagon?

Yes. A slanted cut through the middle of the cube that crosses all six faces makes a hexagon. With the right angle, it is a regular hexagon.

Summary

A cross section is the shape where a plane cuts a solid.
Cuts parallel to the base copy the base (smaller for cones and pyramids). Vertical cuts are often rectangles or triangles.
Every cross section of a sphere is a circle.
Spinning a rectangle makes a cylinder, a right triangle makes a cone, and a semicircle makes a sphere.
Use similar triangles for cone and pyramid cuts, and the Pythagorean theorem for sphere cuts.

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