Nets and Cross-Sections of 3D Shapes

There are two powerful ways to understand three-dimensional shapes: unfold them into flat patterns called nets, or slice them with a flat plane to reveal cross-sections. Both concepts appear on the GED and SAT, and both are used daily in trades like sheet metal fabrication and HVAC ductwork. This page covers the most common nets and cross-sections you need to know.

What Is a Net?

A net is a flat, two-dimensional pattern that folds up into a three-dimensional shape. If you took a cardboard box and cut along its edges until it lay flat on a table, the resulting flat shape is a net of a rectangular prism.

The key idea is that the surface area of a 3D solid equals the total area of its net. Every face of the solid appears exactly once in the net, and the faces are connected along shared edges (called fold lines). When you fold along those lines, the flat pattern closes up into the original solid.

Common Nets

Cube

A cube has 6 square faces. The most recognizable net is the cross (or plus) shape — four squares in a column with one square on each side of the second square. However, there are actually 11 distinct nets that fold into a cube. Not every arrangement of six connected squares works — the squares must be positioned so they fold without overlapping.

Rectangular Prism

A rectangular prism (box) has 6 rectangular faces arranged in three matching pairs: top and bottom ($l \times w$), front and back ($l \times h$), and left and right ($w \times h$). When unfolded, the net shows all six faces connected along fold lines.

The dashed lines in the diagram above represent fold lines — the edges where two faces share a boundary. Folding along every dashed line closes the net into a box.

Cylinder

A cylinder has three surfaces: two circular ends and one curved side. When you unroll the curved side, it becomes a rectangle. The width of that rectangle equals the circumference of the circle ($2\pi r$), and the height of the rectangle equals the height of the cylinder ($h$).

This net explains the cylinder surface area formula: the two circles contribute $2\pi r^2$ and the rectangle contributes $2\pi rh$ (circumference times height), giving $SA = 2\pi r^2 + 2\pi rh$.

Cone

A cone has two surfaces: a circular base and a lateral surface. When you unroll the curved side of a cone, it becomes a sector of a larger circle — like a wedge of pie. The radius of that sector equals the cone’s slant height $l$, and the arc length of the sector equals the circumference of the base ($2\pi r$).

Square Pyramid

A square pyramid unfolds into one square (the base) with four triangles attached to its sides. Each triangle has a base equal to the side of the square and a height equal to the slant height of the pyramid.

What Is a Cross-Section?

A cross-section is the two-dimensional shape you get when you slice a three-dimensional solid with a flat plane. Think of it like cutting a loaf of bread — every slice reveals a flat shape. The shape of the cross-section depends on two things: the solid being sliced and the angle of the cut.

Cross-sections are important in architecture, engineering, medical imaging (CT scans are literally cross-sections of the body), and manufacturing. On the GED and SAT, you will be asked to identify what shape appears when a given solid is sliced at a specific angle.

Cross-Sections of Common Solids

Rectangular Prism

• Slice parallel to a face: produces a rectangle (same shape as that face)
• Slice perpendicular to the length: produces a rectangle (same shape as the end face)
• Diagonal slice: depending on the angle, can produce a rectangle, parallelogram, triangle (if the plane passes through an edge and opposite face), or even a hexagon (if the plane passes through all six faces)

Cylinder

• Slice parallel to the base: produces a circle with the same radius
• Slice perpendicular to the base (through the center): produces a rectangle with width $2r$ and height $h$
• Slice at an angle: produces an ellipse — a stretched circle

Cone

The cross-sections of a cone are especially important because they form the four conic sections studied in higher mathematics:

• Parallel to the base: produces a circle (smaller than the base, with radius proportional to the distance from the apex)
• Through the apex, perpendicular to the base: produces a triangle (specifically, an isosceles triangle)
• At an angle, not parallel to the base or slant side: produces an ellipse
• Parallel to the slant edge: produces a parabola

The conic sections (circle, ellipse, and parabola) all come from slicing a single cone. A hyperbola requires slicing a double cone (two cones joined at their apex). All four conic sections are studied more deeply in precalculus.

Sphere

• Every cross-section of a sphere is a circle
• The largest possible cross-section passes through the center of the sphere — this is called a great circle, and its radius equals the radius of the sphere
• Cross-sections taken farther from the center produce smaller circles
• If the slicing plane is a distance $d$ from the center, the cross-section radius is $r_{\text{cross}} = \sqrt{r^2 - d^2}$

Pyramid (Square Base)

• Slice parallel to the base: produces a smaller square (similar to the base, scaled down proportionally)
• Slice perpendicular to the base through the apex: produces a triangle (specifically, an isosceles triangle)
• Slice perpendicular to the base but not through the apex: produces a trapezoid

Worked Examples

Example 1: Identifying a valid cube net

A flat pattern shows six connected squares arranged in an L-shape: four squares in a row, with one square below the left end and one square below the right end. Does this fold into a cube?

Solution: Visualize the folding. The four squares in a row would form four faces wrapping around the cube. The square below the left end becomes the bottom, and the square below the right end would need to become the top. However, when you fold the row of four squares into a loop, the two extra squares end up on the same side of the cube, overlapping. This arrangement does not fold into a cube.

Answer: No, this is not a valid cube net.

Example 2: Dimensions of a cylinder’s net

A cylinder has radius $r = 4$ cm and height $h = 10$ cm. What are the dimensions of the rectangular part of its net?

Solution: The rectangular part of the net represents the lateral surface of the cylinder. When unrolled:

• Width = circumference of the base = $2\pi r = 2\pi(4) = 8\pi \approx 25.13$ cm
• Height = height of the cylinder = $10$ cm

Answer: The rectangle is approximately 25.13 cm wide and 10 cm tall.

Example 3: Cross-section of a cylinder sliced parallel to its base

A cylinder with radius 6 cm and height 15 cm is sliced with a plane that is parallel to its base and 5 cm above the bottom. What is the shape and size of the cross-section?

Solution: When you slice a cylinder parallel to its base, the cross-section is always a circle with the same radius as the base, regardless of where you make the cut. The height above the bottom does not matter — every horizontal slice through a cylinder looks the same.

Answer: The cross-section is a circle with radius 6 cm.

Example 4: Cross-section of a cone sliced at half its height

A cone has a base radius of 8 cm and a height of 12 cm. If you slice it parallel to the base at a height of 6 cm (halfway up), what is the radius of the cross-section?

Solution: When you slice a cone parallel to the base, the resulting circle is similar to the base. The ratio of the cross-section radius to the base radius equals the ratio of the distance from the apex to the cut height versus the total height.

The slice is at 6 cm above the base, which means it is $12 - 6 = 6$ cm below the apex — exactly halfway from the apex to the base.

By similar triangles:

$\frac{r_{\text{cross}}}{r_{\text{base}}} = \frac{\text{distance from apex to cut}}{\text{total height}} = \frac{6}{12} = \frac{1}{2}$

$r_{\text{cross}} = \frac{1}{2} \times 8 = 4 \text{ cm}$

Answer: The cross-section is a circle with radius 4 cm.

Example 5: HVAC application — angled cut through a round duct

An HVAC technician needs to cut a rectangular hole in a wall for a round duct (8-inch diameter) to pass through at a 30-degree angle. What shape is the opening in the wall, and how should the tech size it?

Solution: When a cylinder passes through a flat surface at an angle, the cross-section is an ellipse. The minor axis of the ellipse equals the diameter of the cylinder (the duct just barely fits in that direction), and the major axis is longer because the duct crosses the wall at an angle.

The minor axis is the diameter: $\text{minor axis} = 8$ in.

The major axis depends on the angle. For a cylinder tilted at angle $\theta$ from perpendicular to the wall:

$\text{major axis} = \frac{\text{diameter}}{\cos \theta} = \frac{8}{\cos 30°} = \frac{8}{0.866} \approx 9.24 \text{ in}$

Answer: The opening is an ellipse approximately 9.24 inches tall and 8 inches wide. The technician should cut an elliptical (or slightly oversized rectangular) hole to accommodate the angled duct.

Real-World Application: Sheet Metal Fabrication

In HVAC and sheet metal work, technicians routinely unfold 3D duct shapes into flat patterns — they are literally creating nets. A round duct section is cut from a flat rectangle of sheet metal (width = circumference, height = duct length), then rolled into a cylinder and seamed. Rectangular duct transitions, elbows, and tapered fittings all require the fabricator to calculate the net before cutting.

Understanding nets directly translates to:

• Material estimation — the net area tells you exactly how much sheet metal is needed
• Waste reduction — efficient nesting of net patterns on a metal sheet minimizes scrap
• Accurate fabrication — the net must be mathematically precise or the 3D shape will not close properly

Cross-sections matter too: when a round duct enters a wall at an angle, the hole in the wall is an ellipse, not a circle. Sizing that hole correctly requires understanding cross-sections.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A net is a flat pattern that folds into a 3D shape — the net’s total area equals the solid’s surface area
• A cube has 11 distinct nets, the most common being the cross (plus) shape
• A cylinder’s net is 2 circles + 1 rectangle (width = circumference = $2\pi r$, height = $h$)
• A cone’s net is 1 circle + 1 sector (the curved surface unrolls into a sector with radius equal to the slant height)
• A cross-section is the 2D shape revealed when you slice a solid with a flat plane
• Slicing a cone at different angles produces three conic sections: circle, ellipse, and parabola (a hyperbola requires a double cone)
• Every cross-section of a sphere is a circle; the largest is the great circle through the center
• For cones and pyramids, a slice parallel to the base produces a shape similar to the base, scaled by the ratio of the distance from the apex to the total height
• In HVAC and sheet metal work, nets are used to lay out flat patterns for ductwork, and cross-sections determine the shape of wall openings for angled ducts

Return to Geometry for more topics in this section.