Nets Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Nets.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A net is a two-dimensional layout of all the faces of a three-dimensional solid, arranged so that folding along the edges produces the original solid. Nets reveal the surface area as the sum of flat face areas.

Unfold a 3D solid like a cardboard box—the flat connected pattern you get is a net of that solid.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A net is the flattened layout of all a solid's faces that folds back into the original 3D shape.

Common stuck point: The procedure for nets is the easy part; the trap is drawing a layout that cannot fold into the solid. Asking "Is this a flat layout of all a solid's faces that folds along edges back into the solid?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this a flat layout of all a solid's faces that folds along edges back into the solid?

Worked Examples

Example 1

easy
Draw and describe the net of a rectangular prism (box) with dimensions 55 cm ×3\times 3 cm ×2\times 2 cm, and use the net to find the total surface area.

Answer

Total surface area =62= 62 cm²

First step

1
Step 1: A rectangular prism has 6 faces arranged in 3 pairs of congruent rectangles: two 5×35 \times 3 faces (top/bottom), two 5×25 \times 2 faces (front/back), and two 3×23 \times 2 faces (left/right).

Full solution

  1. 2
    Step 2: The net is obtained by unfolding: lay the bottom flat, unfold the four side faces out, and place the top opposite the bottom. It forms a cross or T-shape with 6 rectangles.
  2. 3
    Step 3: Calculate each pair's area: 2(5×3)=302(5 \times 3) = 30 cm², 2(5×2)=202(5 \times 2) = 20 cm², 2(3×2)=122(3 \times 2) = 12 cm².
  3. 4
    Step 4: Total surface area =30+20+12=62= 30 + 20 + 12 = 62 cm².
A net is the 2D unfolding of a 3D figure. For a rectangular prism, the net has 6 rectangles. Adding their areas (which is the total surface area) gives 62 cm². Nets make surface area calculation visually intuitive.

Example 2

medium
Describe the net of a regular triangular pyramid (tetrahedron with equilateral triangle faces) with edge length 66 cm. Find its total surface area using the net.

Example 3

medium
A square pyramid has a base of side 88 and triangular faces with slant height 55. Use its net to find the surface area.

Example 4

medium
A triangular prism has equilateral triangular bases of side 66 and length 1010. Find the total surface area from its net.

Example 5

hard
A pentagonal pyramid has a base that is a regular pentagon with side 44 and apothem ≈2.75\approx 2.75, and triangular faces with slant height 66. Estimate the surface area from its net.

Example 6

challenge
A right cone has base radius rr and slant height ℓ\ell. Show that the lateral surface area in its net is πrℓ\pi r \ell.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
How many faces does a triangular prism have, and what shapes are they? If the triangular base has base 44 cm and height 33 cm, and the prism length is 1010 cm, find the surface area using a net.

Example 2

hard
A cylindrical can has radius 33 cm and height 1010 cm. Describe its net and find the total surface area.

Example 3

easy
What 3D solid does a net consisting of 4 triangles fold into?

Example 4

easy
True or false: every arrangement of 6 squares is a valid net of a cube.

Example 5

easy
What solid has a net of one rectangle and one rectangle of width 2πr2\pi r wrapping around it? (List the missing pieces.)

Example 6

medium
A cylinder has radius 22 and height 55. Find the total surface area in terms of π\pi using its net.

Example 7

medium
Describe the net of a cone with base radius rr and slant height ℓ\ell.

Example 8

medium
Identify the solid whose net is one regular hexagon and six rectangles (each sharing one edge with the hexagon).

Example 9

medium
A net of a rectangular box has total area 9494 cm². If two of its three rectangle types contribute 6060 cm² combined (i.e. one pair plus one face of another type? No — each pair sums shown), find the area of the missing pair when one pair is 3030 cm² and another pair is 4040 cm².

Example 10

hard
A box measures 3×4×53 \times 4 \times 5 cm. An ant on a corner walks across two adjacent faces to reach the diagonally opposite corner. Using a net (unfolding the two faces flat), find the shortest distance.

Example 11

hard
For a cone with base radius 33 and slant height 99, the lateral net is a sector. Find the central angle of that sector (in degrees).

Example 12

hard
A right triangular prism has bases that are right triangles with legs 33 and 44, and prism length 1010. Find its surface area using a net.

Example 13

challenge
A box measures 1×2×31 \times 2 \times 3. An ant walks from (0,0,0)(0,0,0) to (1,2,3)(1,2,3) along the surface. Find the shortest such distance.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

surface areashapescross sections 3d