Grade: Grade 6 Subject: Mathematics Unit: Geometry SAT: Geometry+Trigonometry ACT: Math

Surface Area

Learn how to calculate the total area covering the outside of 3D shapes using nets and formulas.

What is Surface Area?

Surface Area = Total Area of ALL Faces

Surface area is the total area that covers the entire outside of a 3D shape. Think of it as the amount of wrapping paper needed to completely cover a box!

While area measures flat 2D surfaces, surface area measures the outside of 3D objects. Every 3D shape has faces (flat surfaces), and we add up the area of each face to find the total surface area.

Key Ideas About Surface Area

  • Surface area is measured in square units (cm², m², in², ft²)
  • Count all faces - don't forget the top, bottom, and all sides
  • Opposite faces are often equal - this helps simplify calculations
  • Nets help visualize - unfolding a 3D shape shows all faces

Why is Surface Area Useful?

🎁

Wrapping Gifts

Calculate how much wrapping paper you need for a box

🎨

Painting

Determine how much paint to cover a room or object

📦

Manufacturing

Calculate material needed to make boxes or containers

🏗️

Construction

Figure out siding, roofing, or flooring materials

Understanding Nets

A net is what you get when you "unfold" a 3D shape and lay it flat. Nets help us see all the faces at once, making it easier to calculate surface area.

Rectangular Prism and Its Net

3D Rectangular Prism
l w h

l = length, w = width, h = height

Unfolded Net
Top Front Bottom Back Left Right

All 6 faces laid flat

Surface Area of a Rectangular Prism

A rectangular prism (like a box) has 6 faces that come in 3 pairs of equal rectangles:

  • Top and Bottom: length x width (2 faces)
  • Front and Back: length x height (2 faces)
  • Left and Right: width x height (2 faces)

Surface Area of a Rectangular Prism

SA = 2lw + 2lh + 2wh

Or equivalently: SA = 2(lw + lh + wh)

l = length, w = width, h = height

Surface Area of a Cube

A cube is a special rectangular prism where all edges are equal. Since every face is a square with side length s:

Cube and Its Net

3D Cube
s

All sides equal length s

Cube Net (Cross Shape)
s x s s x s

6 identical square faces

Surface Area of a Cube

SA = 6s²

s = side length

Since all 6 faces are squares with area s²

Pro Tip: When calculating surface area, organize your work by listing each face and its area. Then add them all up. This helps you avoid missing any faces!

Worked Examples

Let's work through some surface area problems step by step.

Example 1: Surface Area of a Rectangular Prism

Find the surface area of a box that is 5 cm long, 3 cm wide, and 4 cm tall.
1
Identify the dimensions:
Length (l) = 5 cm, Width (w) = 3 cm, Height (h) = 4 cm
2
Calculate the area of each pair of faces:
Top & Bottom (lw): 5 x 3 = 15 cm² (x2 = 30 cm²)
Front & Back (lh): 5 x 4 = 20 cm² (x2 = 40 cm²)
Left & Right (wh): 3 x 4 = 12 cm² (x2 = 24 cm²)
3
Add all the face areas together:
SA = 30 + 40 + 24 = 94 cm²
Answer: The surface area is 94 cm²

Example 2: Surface Area of a Cube

A Rubik's cube has sides that measure 5.7 cm. What is the total surface area?
1
Identify the side length:
s = 5.7 cm
2
Use the cube surface area formula:
SA = 6s²
3
Substitute and calculate:
SA = 6 x (5.7)²
SA = 6 x 32.49
SA = 194.94 cm²
Answer: The surface area is 194.94 cm²

Example 3: Real-World Application - Wrapping Paper

You need to wrap a gift box that measures 12 inches long, 8 inches wide, and 4 inches tall. How much wrapping paper do you need (minimum)?
1
Set up the problem:
l = 12 in, w = 8 in, h = 4 in
2
Use the formula SA = 2(lw + lh + wh):
SA = 2(12x8 + 12x4 + 8x4)
3
Calculate step by step:
SA = 2(96 + 48 + 32)
SA = 2(176)
SA = 352 in²
Answer: You need at least 352 square inches of wrapping paper

Example 4: Painting a Room

A storage container is 10 ft long, 6 ft wide, and 8 ft tall. If one can of paint covers 100 ft², how many cans do you need to paint the outside (not the bottom)?
1
Identify what we're painting (5 faces, no bottom):
Top: lw = 10 x 6 = 60 ft²
Front & Back: lh = 10 x 8 = 80 ft² each (160 ft² total)
Left & Right: wh = 6 x 8 = 48 ft² each (96 ft² total)
2
Add the areas:
SA = 60 + 160 + 96 = 316 ft²
3
Calculate paint cans needed:
316 ÷ 100 = 3.16 cans
Since you can't buy partial cans, round up!
Answer: You need 4 cans of paint

Practice Problems

Try these problems on your own. Click on the correct answer!

Problem 1: Basic Rectangular Prism

Find the surface area of a rectangular prism with length = 6 cm, width = 4 cm, and height = 2 cm.

6 cm x 4 cm x 2 cm

Problem 2: Cube Surface Area

What is the surface area of a cube with side length 7 inches?

Problem 3: Real-World Application

A cereal box is 30 cm tall, 20 cm wide, and 5 cm deep. What is the total surface area of the box?

Problem 4: Finding a Dimension

A cube has a surface area of 150 ft². What is the length of one side?

Check Your Understanding: Surface Area Challenge

Test your surface area skills with this 6-question challenge!

Surface Area Challenge

Score: 0 / 6
Question 1 of 6

Challenge Complete!

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Next Steps

Key Takeaways:

  • Surface area is the total area of all faces of a 3D shape
  • Rectangular prism: SA = 2lw + 2lh + 2wh (or 2(lw + lh + wh))
  • Cube: SA = 6s² (six identical square faces)
  • Nets help visualize all faces by unfolding the 3D shape
  • Always include units squared (cm², m², in², ft²) in your answer
  • Practice finding surface area of objects around your home
  • Try sketching nets for different boxes you see
  • Review area formulas for rectangles and squares
  • Return to the unit page for more geometry topics