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Find Surface Area
Using Nets

Learn to unfold 3D shapes into flat patterns called nets, then calculate the total surface area by adding up every face.

6th Grade Math

What Is Surface Area?

Surface area is the total area of all the outer faces of a three-dimensional figure. Imagine you could wrap a gift box in paper: the amount of paper you need equals the surface area of that box.

What Is a Net?

A net is a flat, two-dimensional pattern that can be folded up to form a three-dimensional shape. When you "unfold" a 3D figure, you lay all its faces flat, and that flat layout is the net.

Top Front Left Right Back Bottom

Why Use Nets for Surface Area?

See Every Face
A net lays out all faces flat so you never miss a hidden side.
Find Each Area
Each face is a simple rectangle or triangle whose area you already know how to find.
Add Them Up
Total surface area \( = \) sum of the areas of all faces in the net.

The Surface Area Formula

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

This comes from adding the areas of all 6 rectangular faces shown in the net.

Net of a Rectangular Prism

\( l \times w \)
\( l \times h \)
\( w \times h \)
\( w \times h \)
\( l \times w \)
\( l \times h \)
 6 faces: 3 pairs of matching rectangles

Net of a Triangular Prism

Side 1 Base Side 2 Triangle Triangle 5 faces total: 3 rectangles 2 triangles

Surface Area vs Volume

Surface Area Volume
Measures Total area of the outside Space inside the figure
Units Square units (\(\text{cm}^2\)) Cubic units (\(\text{cm}^3\))
Think of it as Wrapping paper needed Water it can hold
Found with Nets (add face areas) \( l \times w \times h \)

How to Find Surface Area

1
Identify the 3D shape. Is it a rectangular prism, triangular prism, cube, or pyramid?
2
Draw or picture the net. Unfold the shape in your mind and sketch each face flat.
3
Label dimensions. Write the length and width (or base and height) on every face.
4
Calculate each face's area. Use \( A = lw \) for rectangles and \( A = \frac{1}{2}bh \) for triangles.
5
Add all areas together. The sum is the total surface area in square units.

Example 1: Rectangular Prism

Find the surface area of a box with \( l = \) 5 cm, \( w = \) 3 cm, \( h = \) 4 cm.

5 cm 4 cm 3 cm Calculation:
Top/Bottom: \( 2(5 \times 3) = 30 \)
Front/Back: \( 2(5 \times 4) = 40 \)
Left/Right: \( 2(3 \times 4) = 24 \)
\( \text{SA} = 94 \text{ cm}^2 \)

Example 2: Cube

Find the surface area of a cube with side length \( s = \) 6 in.

6 6 6 6 6 6 All 6 faces are squares:
\( \text{Each face} = 6 \times 6 = 36 \)
\( \text{SA} = 6 \times 36 = 216 \)
\( \text{SA} = 216 \text{ in}^2 \)

Example 3: Triangular Prism

A triangular prism has triangle base \( b = \) 6 cm, triangle height \( h = \) 4 cm, side lengths 5 cm, and prism length 10 cm.

\( 5 \times 10 \)
\( 6 \times 10 \)
\( 5 \times 10 \)
 triangle triangle Faces:
2 triangles: \( 2(\frac{1}{2} \!\times\! 6 \!\times\! 4) = 24 \)
Base rect: \( 6 \times 10 = 60 \)
Side rects: \( 2(5 \times 10) = 100 \)
\( \text{SA} = 184 \text{ cm}^2 \)

Example 4: Triangular Pyramid

A triangular pyramid has a base with sides 8 m and base height 6.93 m. Each triangular face has base 8 m and slant height 10 m.

Base Face 1 Face 2 Face 3 Faces:
Base: \( \frac{1}{2}(8)(6.93) = 27.72 \)
3 faces: \( 3(\frac{1}{2} \!\times\! 8 \!\times\! 10) = 120 \)
\( \text{SA} = 147.72 \text{ m}^2 \)

Common Mistakes to Avoid

✕ Forgetting a Face
A rectangular prism has 6 faces, not 4 or 5. Draw the net to make sure you count every one.
✕ Using Wrong Units
Surface area uses square units (\(\text{cm}^2\)), not plain units (\(\text{cm}\)). Always include the "squared" symbol.
✕ Confusing Height
For triangles, use the perpendicular height of the triangle, not the slant edge of the 3D shape, unless specified.

Summary of Surface Area Formulas

Shape Faces in Net Surface Area Formula
Cube 6 squares \( 6s^2 \)
Rectangular Prism 6 rectangles (3 pairs) \( 2lw + 2lh + 2wh \)
Triangular Prism 3 rectangles + 2 triangles \( bh + l(a+b+c) \) *
Triangular Pyramid 4 triangles Sum of 4 triangle areas

Question 1 of 5

A cube has a side length of 4 cm. What is its surface area?

Question 2 of 5

A rectangular prism has \( l = 8 \), \( w = 3 \), and \( h = 5 \). What is the surface area?

Question 3 of 5

How many faces does a triangular prism have in its net?

Question 4 of 5

A triangular prism has triangular bases with \( b = 6 \) and \( h = 4 \), and a prism length of \( 10 \). The triangle sides are \( 5, 6, 5 \). What is the surface area?

Question 5 of 5

You want to wrap a gift box that is 12 in long, 8 in wide, and 6 in tall. How much wrapping paper do you need at minimum?