Surface Area of Rectangular Prisms

Learn how to calculate the total surface area of rectangular prisms using a simple formula.

Intermediate25 minLesson
Printable worksheet

Definition

The surface area of a 3D shape is the total area of all its outer faces combined.
A rectangular prism (also called a cuboid or box) has 6 rectangular faces:
  • 2 faces with dimensions l×wl \times w (top and bottom)
  • 2 faces with dimensions l×hl \times h (front and back)
  • 2 faces with dimensions w×hw \times h (left and right)
Surface Area=2(lw+lh+wh)\text{Surface Area} = 2(lw + lh + wh)
Where:
  • ll = length
  • ww = width
  • hh = height

Try it now

A rectangular prism has 6 faces. How do they come in groups?

Worked Examples

A gift box has length 8 cm, width 5 cm, and height 3 cm. Find its surface area.
1

Identify the dimensions

l=8l = 8 cm, w=5w = 5 cm, h=3h = 3 cm → Three dimensions identified
2

Calculate each pair of faces

lw=8×5=40lw = 8 \times 5 = 40 sq cm lh=8×3=24lh = 8 \times 3 = 24 sq cm wh=5×3=15wh = 5 \times 3 = 15 sq cm → Three products found
3

Add the three products

40+24+15=7940 + 24 + 15 = 79 sq cm → Sum of face areas
4

Multiply by 2 (for paired faces)

2×79=1582 \times 79 = 158 sq cm → Total surface area

Common Mistakes

Forgetting to multiply by 2

Why it's wrong: Students calculate lw+lh+whlw + lh + wh and stop, forgetting that each dimension pair creates TWO faces.

Correct: Always remember: a rectangular prism has 6 faces that come in 3 pairs. The final step is multiplying by 2.

Confusing surface area with volume

Why it's wrong: Both use length, width, and height, but volume (V=lwhV = lwh) is a single multiplication, while surface area requires adding and doubling.

Correct: Surface area measures the outside covering (2D, in square units). Volume measures inside space (3D, in cubic units).

Using wrong units

Why it's wrong: Surface area is measured in square units, not regular units.

Correct: Always write sq cm, sq m, or use the superscript: cm2^2, m2^2

Interactive Visual

3D Shape Viewer

Faces

6

Edges

12

Vertices

8

Euler's formula: V − E + F = 8 − 12 + 6 = 2

Volume

V = lwh

V = 8 · 5 · 3 = 120

120 units³

Surface Area

SA = 2(lw + lh + wh)

SA = 2(8 · 5 + 8 · 3 + 5 · 3) = 158

158 units²

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

16 problems
Problem 1 of 16
Easy
A rectangular prism has 6 faces. How do they come in groups?

Why It Matters

Surface area calculations are essential in everyday life and many careers:
  • Wrapping presents: How much wrapping paper do you need?
  • Painting rooms: How much paint to cover all walls?
  • Manufacturing: How much material to make a cardboard box?
  • Construction: How many tiles to cover a swimming pool?
  • Packaging design: Minimizing material costs while maximizing volume
Understanding surface area helps you solve practical problems and make informed decisions about materials and costs.

Real World Applications

Wrapping Paper

Gift shops and crafters need to know exactly how much wrapping paper to cut for each box size.

Example:

A shoebox measuring 30 cm by 20 cm by 12 cm needs wrapping paper. SA = 2(30(20) + 30(12) + 20(12)) = 2(600 + 360 + 240) = 2(1200) = 2400 sq cm of paper.

1Try It Yourself

You have a gift box that is 25 cm long, 15 cm wide, and 10 cm high.

How much wrapping paper do you need?

Step 1: Write the mathematical expression

Calculate: 2(25×15+25×10+15×10)2(25 \times 15 + 25 \times 10 + 15 \times 10)

Construction and Painting

Painters calculate wall areas to estimate how much paint to buy for a room.

Example:

A room 4 m by 5 m by 3 m high has 4 walls. The wall area (not including floor and ceiling) is 2(4 + 5) times 3 = 54 sq m.

2Try It Yourself

An aquarium tank is 80 cm long, 40 cm wide, and 50 cm high. You need to paint the outside of all 6 faces with waterproof sealant.

What is the total surface area to paint?

Step 1: Write the mathematical expression

Calculate: 2(80×40+80×50+40×50)2(80 \times 40 + 80 \times 50 + 40 \times 50)

Key Takeaways

  • 1Surface area is the total area of all faces of a 3D shape
  • 2A rectangular prism has 6 faces in 3 pairs (top/bottom, front/back, left/right)
  • 3Formula: SA=2(lw+lh+wh)SA = 2(lw + lh + wh)
  • 4Surface area is measured in square units (sq cm, sq m, cm2^2, m2^2)
  • 5Remember to multiply by 2 because faces come in pairs

Frequently Asked Questions

Surface area measures the outside covering of a 3D shape (like wrapping paper around a box) and is measured in square units. Volume measures the space inside the shape (how much it can hold) and is measured in cubic units.
Surface area measures the outside covering of a 3D shape (like wrapping paper around a box) and is measured in square units. Volume measures the space inside the shape (how much it can hold) and is measured in cubic units.
A rectangular prism has 6 faces that come in matching pairs: top and bottom (lwlw), front and back (lhlh), left and right (whwh). Multiplying by 2 accounts for both faces in each pair.
For a cube with side ss, the formula simplifies to SA=6s2SA = 6s^2 because all 6 faces are identical squares.

Glossary

Surface area
The total area of all outer faces of a 3D shape, measured in square units
Rectangular prism
A 3D shape with 6 rectangular faces (also called a cuboid or box)
Face
A flat surface on a 3D shape
Dimension
A measurable extent such as length, width, or height
Square units
Units for measuring area, such as cm2^2, m2^2, or square feet
Today's challengeOne problem, three tries

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