Surface Area of a Right Prism

Statement

The surface area of a right prism is the total area of all its faces. A right prism has two identical parallel cross-sections (front and back) and rectangular faces joining them. The general formula is:

\[ S = \text{perimeter of cross-section} \times \text{length} + 2 \times \text{area of cross-section} \]

The first part \(\text{perimeter} \times \text{length}\) gives the total area of the rectangular side faces. The second part \(2 \times \text{area of cross-section}\) accounts for the front and back ends.

Why it’s true

• A prism is built by extending a 2D shape (cross-section) along a straight length.
• The two ends are congruent copies of the cross-section, giving \(2 \times \text{area of cross-section}\).
• The side faces are rectangles. Each side of the cross-section, when stretched through the prism’s length, creates a rectangle of area “side × length”. Adding all these rectangles together gives \(\text{perimeter} \times \text{length}\).
• Adding both contributions gives the total surface area.

Recipe (how to use it)

1. Identify the shape of the cross-section (triangle, trapezium, hexagon, etc.).
2. Calculate its perimeter.
3. Calculate its area.
4. Multiply perimeter by the prism’s length.
5. Add \(2 \times \text{area of cross-section}\).
6. Label the final answer with square units.

Spotting it

Use this formula whenever you have a prism (a solid with identical parallel cross-sections along its length). Common exam prisms include triangular, trapezoidal, or hexagonal prisms.

Common pairings

• Volume of a prism: \(V = \text{area of cross-section} \times \text{length}\).
• 2D area and perimeter formulas for polygons and circles.

Mini examples

1. Given: Triangular prism with base triangle area \(12\,cm^2\), perimeter \(18\,cm\), length \(10\,cm\). Answer: \(S = 18 \times 10 + 2 \times 12 = 180 + 24 = 204\,cm^2\).
2. Given: Rectangular prism (cross-section 6 by 4). Area=24, Perimeter=20, length=15. Answer: \(S = 20 \times 15 + 2 \times 24 = 300 + 48 = 348\,cm^2\).

Pitfalls

• Forgetting to double the cross-section area.
• Mixing volume with surface area.
• Calculating perimeter incorrectly when the cross-section is irregular.

Exam strategy

• Write the cross-section dimensions clearly and check your perimeter and area.
• Always separate the two parts of the formula before adding.
• If it’s a triangular prism, be especially careful with triangle area (½ base × height).

Summary

Right prisms are solids formed by extending a 2D shape along a length. To find the surface area, calculate all side rectangles (\(\text{perimeter} \times \text{length}\)) and the two ends (\(2 \times \text{area of cross-section}\)). This formula is essential for GCSE questions involving packaging, engineering, and 3D modelling.