Surface Area of Prisms
Unfold 3D prisms into 2D nets, calculate every face, and solve real-world packaging and painting problems.
Printable Worksheets
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Worksheet
Download or print the worksheet to work through this lesson.
Q1: Imagine unfolding a shoebox into a flat shape. How many rectangles would you see? Sketch the net mentally and label the dimensions if the box is 30 cm by 20 cm by 10 cm.
Q2: An open-top box has 5 faces instead of 6. If a closed box needs 600 cm$^2$ of cardboard, how much less would an open-top box need? Explain your reasoning.
Learning Intentions
Know
- That surface area is the total area of all faces of a 3D object.
- The formulas for surface area of rectangular and triangular prisms.
Understand
- How a net represents every face of a prism.
- Why open prisms have one fewer face than closed prisms.
Can Do
- Calculate the surface area of a prism from its net or dimensions.
- Solve practical problems involving paint, packaging and material costs.
Success Criteria
- I can draw the net of a rectangular or triangular prism.
- I can calculate the surface area of a closed prism by adding all face areas.
- I can adjust my calculation for an open prism (missing top, bottom or side).
Key Terms
Common Mistakes to Avoid
Wrong: Counting only the visible faces in a 3D sketch. The bottom face and back faces are also part of the surface area.
Right: A rectangular prism has 6 faces: top, bottom, front, back, left, right. Count them all.
Wrong: Using the slant height instead of the perpendicular height for the rectangular faces of a triangular prism.
Right: The lateral faces of a prism always use the perpendicular height of the prism, not the slant height of the triangular base.
A net is what you get when you unfold a 3D shape. Every face appears exactly once.
For a rectangular prism, the net consists of 6 rectangles arranged in a cross or T-shape:
- Top and bottom (identical)
- Front and back (identical)
- Left and right (identical)
For a triangular prism, the net consists of:
- Two triangular bases (identical)
- Three rectangular lateral faces
What to write in your book
- A rectangular prism has 6 faces: top, bottom, front, back, left, right
- Net: unfold the prism so every face appears exactly once
- Formula: $SA = 2(lw + lh + wh)$
- Opposite faces are identical in area
Correct! $SA = 2(5\times4 + 5\times3 + 4\times3) = 2(20+15+12) = 2\times47 = 94$ cm$^2$.
Not quite. $SA = 2(5\times4 + 5\times3 + 4\times3) = 2(20+15+12) = 2\times47 = 94$ cm$^2$.
An open prism is missing one or more faces. You must subtract the area of any missing face from the total.
Common open prism scenarios:
- Open-top box: Missing the top face. Subtract $lw$ from total SA.
- Open-ended pipe: Missing both circular ends (cylinder). Subtract $2 \times \pi r^2$.
- Display case with open front: Missing one rectangular face. Subtract that face area.
What to write in your book
- An open prism is missing one or more faces
- Open-top box: subtract $lw$ from total SA
- Always identify which face(s) are missing before calculating
Correct! Closed SA = $2(10\times8 + 10\times6 + 8\times6) = 376$ cm$^2$. Subtract top = $10\times8 = 80$. Open SA = $376 - 80 = 296$ cm$^2$.
Not quite. Closed SA = $2(10\times8 + 10\times6 + 8\times6) = 376$ cm$^2$. Subtract top = $10\times8 = 80$. Open SA = $376 - 80 = 296$ cm$^2$.
Interactive: Net Unfolder
Your Turn
Question 1: Find the surface area of a closed rectangular prism with dimensions 15 cm by 10 cm by 6 cm.
Question 2: A triangular prism has a right-angled triangular base with sides 3 cm, 4 cm and 5 cm. The prism length is 10 cm. Find its total surface area.
Question 3: An open-top fish tank measures 80 cm by 50 cm by 40 cm. Calculate the area of glass needed.
Revisit Your Thinking
Look back at your Think First answer about the shoebox net. How many rectangles did you identify? Using the formula, calculate the exact surface area of the shoebox (30 cm by 20 cm by 10 cm). Did your visualisation match the actual net? Explain any differences.
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
A closed rectangular prism has dimensions 5 cm, 4 cm and 3 cm. What is its surface area?
An open-top box has dimensions 10 cm by 8 cm by 6 cm. What is its surface area?
A triangular prism has a base triangle with base 6 cm and height 4 cm. The prism length is 10 cm. What is the area of the two triangular ends?
Which of the following is NOT a face of a closed rectangular prism?
A paint tin states that 1 litre covers 12 m$^2$. How many litres are needed to paint the outside of a closed box with surface area 48 m$^2$?
A closed storage container has dimensions 2.5 m by 1.8 m by 1.2 m.
(a) Calculate the total surface area. (2 marks)
(b) If the container is open at the top, what is the new surface area? (1 mark)
A triangular prism tent has a right-angled triangular cross-section with sides 2.4 m, 1.8 m and 3.0 m. The tent is 3.5 m long.
(a) Find the area of the two triangular ends. (1 mark)
(b) Find the total area of the three rectangular sides. (2 marks)
(c) Find the total surface area of the closed tent. (1 mark)
A company manufactures open-top cardboard boxes for shipping. Each box measures 40 cm by 30 cm by 25 cm. The cardboard costs $\$3.50$ per m$^2$.
(a) Calculate the area of cardboard needed for one box. (2 marks)
(b) Calculate the cost of cardboard for one box. (1 mark)
(c) The company makes 500 boxes. Calculate the total cardboard cost. (1 mark)
(d) Explain why the company might choose to make the boxes open-top rather than closed. (1 mark)
Surface Area
Total area of all faces
Rectangular Prism
$SA = 2(lw+lh+wh)$
Triangular Prism
$SA = 2A_{base} + P \times L$
Open Prism
Subtract missing face(s)
Net
2D unfolded pattern
6 Faces
Top, bottom, 4 sides
Real-Life Link
Every package that arrives at your door has been engineered to minimise surface area (to save cardboard) while maximising volume (to fit more product). Australian packaging standards require companies to optimise box dimensions to reduce waste. When painting a house, painters calculate the surface area of walls, ceilings and trim to quote accurately and order the right amount of paint. A typical Australian room might need 15-20 m$^2$ of paint per coat. Understanding surface area helps consumers verify quotes and avoid being overcharged for materials.
Game Time!
Test your surface area skills in an interactive challenge.
Play Surface Area Challenge