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Surface Area
with Nets

Learn to unfold 3D shapes into flat nets, identify every face, and calculate total surface area of prisms and pyramids.

Pre-Algebra

What Is Surface Area?

Surface area is the total area of all the faces (flat surfaces) that cover the outside of a 3D shape. Think of it as the amount of wrapping paper you would need to completely cover a box with no overlaps.

What Is a Net?

A net is what you get when you unfold a 3D shape and lay it completely flat. Every face of the solid becomes a 2D shape you can see, measure, and add up.

3D Box unfold Net (unfolded) 6 faces visible

Why Do Nets Matter?

See Every Face
Nets make hidden faces visible so you never miss one in your calculation.
Use 2D Formulas
Each face is a rectangle, triangle, or other flat shape whose area formula you already know.
Real-World Uses
Packaging, painting walls, wrapping gifts: all require knowing surface area.

The Surface Area Formula

For a rectangular prism with length \(l\), width \(w\), and height \(h\):

Net of a Rectangular Prism

Top \( l \times w \) Front \( l \times h \) Bottom \( l \times w \) Left \( w \times h \) Right \( w \times h \) Back \( l \times h \) \( h \) \( l \) \( w \) 6 faces: 3 pairs of matching rectangles

Net of a Triangular Prism

Side 1 Side 2 Side 3 Base Base 5 faces: 2 triangles (bases) 3 rectangles (sides) Triangular prism net: 2 triangles + 3 rectangles

Net of a Square Pyramid

Base Face Face Face Face 5 faces: 1 square base 4 triangles Slant height is used for the triangular faces

Prism vs. Pyramid

Prism Pyramid
Bases 2 identical polygons 1 polygon
Lateral faces Rectangles Triangles
Comes to a point? No Yes (apex)
Height used Height of prism Slant height of face
Total faces \(n + 2\) for \(n\)-gon base \(n + 1\) for \(n\)-gon base

How to Find Surface Area

1
Identify the solid. Is it a rectangular prism, triangular prism, pyramid, or another shape?
2
Draw or visualize the net. Unfold the shape flat and label every face with its dimensions.
3
Find each face's area. Use \(A = lw\) for rectangles and \(A = \tfrac{1}{2}bh\) for triangles.
4
Add them all up. Sum every face area. Don't forget matching pairs.
5
Write your answer in square units. Example: \(SA = 150 \text{ cm}^2\).

Example 1: Rectangular Prism

8 cm 5 cm 3 cm \( l=8,\; w=3,\; h=5 \) Front/Back: \(8 \times 5 = 40\) Left/Right: \(3 \times 5 = 15\) Top/Bottom: \(8 \times 3 = 24\) \(SA = 2(40+15+24)\) \(SA = 158 \text{ cm}^2\)

Example 2: Triangular Prism

6 cm 5 cm 5 cm 4 cm Triangle base Prism length = 10 cm 2 triangles: \(2 \times \tfrac{1}{2}(6)(4) = 24\) Bottom rect: \(6 \times 10 = 60\) Side rect 1: \(5 \times 10 = 50\) Side rect 2: \(5 \times 10 = 50\) \(SA = 24 + 60 + 50 + 50\) \(SA = 184 \text{ cm}^2\)

Example 3: Square Pyramid

10 m slant = 8 m height Apex \(s = 10\) m, \(l = 8\) m Base: \(10 \times 10 = 100\) 4 triangles: \(4 \times \tfrac{1}{2}(10)(8)\) \(= 4 \times 40 = 160\) \(SA = 100 + 160\) \(SA = 260 \text{ m}^2\)

Why the Net Strategy Works

1
No hidden faces. When you unfold a 3D shape, every single face is visible. You can't accidentally miss the back or bottom.
2
Dimensions become clear. Each face lays flat, so you can directly see its length and width (or base and height for triangles).
3
Easy to organize. You can label and compute each face one at a time, then simply add. No formulas to memorize beyond basic area.

Common Mistakes

Forgetting a Face
Especially the base of a pyramid or the back face of a prism. Always count faces first.
Using Height Instead of Slant Height
Pyramid triangular faces use slant height \(l\), not the vertical height of the pyramid.
Forgetting to Double
Prisms have matching pairs of faces. If you find the front face, remember the back is the same size.

Summary of Formulas

Solid Faces SA Formula
Rectangular Prism 6 rectangles \(2(lw + lh + wh)\)
Triangular Prism 2 triangles + 3 rectangles \(2 \cdot \tfrac{1}{2}bh + P \cdot L\)
Square Pyramid 1 square + 4 triangles \(s^2 + 2sl\)
Triangular Pyramid 4 triangles Sum of 4 triangle areas

Question 1 of 5

A rectangular prism has dimensions \(l = 6\) cm, \(w = 4\) cm, and \(h = 3\) cm. What is the surface area?

Question 2 of 5

A square pyramid has a base side length of \(s = 6\) m and a slant height of \(l = 5\) m. What is the surface area?

Question 3 of 5

How many faces does the net of a triangular prism have?

Question 4 of 5

A cube has a side length of \(s = 7\) in. What is its surface area?

Question 5 of 5

When calculating the surface area of a pyramid, why must you use the slant height rather than the vertical height?