Surface Area of Prisms and Pyramids

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

Surface Area of Rectangular Prisms: Wrapping Paper Math

Imagine you're wrapping a birthday present. How much wrapping paper do you need? You're actually solving a surface area problem — finding the total area of all the faces that cover a 3D shape.

A rectangular prism is like a box — it has 6 flat faces, and each face is a rectangle. Think of a cereal box, a smartphone, or a brick. To find the surface area, we need to calculate the area of each face and add them all together.

The Face Detective Method

Every rectangular prism has three pairs of identical faces:

•Front & Back: length × height
•Left & Right: width × height
•Top & Bottom: length × width

Worked Example: Gift Box

A gift box measures 8 inches long, 5 inches wide, and 3 inches tall. Let's find its surface area:

Front & Back faces: 8 × 3 = 24 sq in (each) → 24 × 2 = 48 sq in
Left & Right faces: 5 × 3 = 15 sq in (each) → 15 × 2 = 30 sq in
Top & Bottom faces: 8 × 5 = 40 sq in (each) → 40 × 2 = 80 sq in

Total Surface Area: 48 + 30 + 80 = 158 square inches

🔍 The Formula Shortcut

Instead of calculating each face separately, mathematicians use this formula:

SA = 2(lw + lh + wh)

This formula automatically doubles each face pair! For our gift box: SA = 2(40 + 24 + 15) = 2(79) = 158 sq in. Same answer, faster calculation.

Real-World Applications

Surface area calculations help manufacturers determine how much material they need for packaging, how much paint to cover a room, or how much fabric to upholster furniture. Construction workers use it to calculate siding for buildings, and even app developers use similar concepts when designing 3D interfaces.

🔑 Key Takeaway

Just like you need to know exactly how much wrapping paper to buy for that perfect present, surface area gives us the precise measurement of every outer surface. Whether you're an architect designing skyscrapers or simply wrapping gifts, you're using the same mathematical principle — measuring what covers the outside.

Sample questions

1. A rectangular prism has dimensions 5 cm, 3 cm, and 4 cm. What is its surface area?

○ 94 cm²

○ 60 cm²

○ 94 cm²

✓ 94 cm²? SA = 2(5×3 + 5×4 + 3×4) = 2(15 + 20 + 12) = 2(47) = 94 cm²

Answer: 94 cm²? SA = 2(5×3 + 5×4 + 3×4) = 2(15 + 20 + 12) = 2(47) = 94 cm² — Surface area = 2(lw + lh + wh) = 2(15 + 20 + 12) = 2 × 47 = 94 cm².

2. Find the surface area of a cube with side length 6 cm.

✓ 216 cm²? SA = 6 × 6² = 6 × 36 = 216 cm²

○ 216 cm²

○ 36 cm²

○ 216 cm²

Answer: 216 cm²? SA = 6 × 6² = 6 × 36 = 216 cm² — A cube has 6 faces, each area = 36 cm², total = 216 cm².

3. A rectangular box is 8 ft long, 5 ft wide, and 3 ft tall. How much cardboard is needed to make the box?

○ 158 ft²

○ 120 ft²

✓ 158 ft²? SA = 2(8×5 + 8×3 + 5×3) = 2(40 + 24 + 15) = 2(79) = 158 ft²

○ 158 ft²

Answer: 158 ft²? SA = 2(8×5 + 8×3 + 5×3) = 2(40 + 24 + 15) = 2(79) = 158 ft² — Surface area formula gives 158 ft².

Skills in this topic

Calculate the surface area of a rectangular prism
Calculate the surface area of a triangular prism
Calculate the surface area of a regular pyramid
Use a 2D net to find the surface area of a 3D figure
Solve real-world problems involving surface area (e.g., painting, wrapping)

Practice 50+ questions on this topic

Unlimited interactive practice, progress tracking, and Nova — your AI tutor. Free to start.

Start learning free →