The surface area of a trapezoidal prism equals the area of the two trapezoidal ends plus the area of the four rectangular side faces. The formula is SA = 2 × ½(a + b)h + (a + b + c + d) × H.
Formula
SA = (a + b)h + (a + b + c + d) × H
Where:
a, b = parallel sides of the trapezoid
c, d = non-parallel sides of the trapezoid
h = height of the trapezoid
H = length (depth) of the prism
Where:
a, b = parallel sides of the trapezoid
c, d = non-parallel sides of the trapezoid
h = height of the trapezoid
H = length (depth) of the prism
Step-by-Step Method
- Find the trapezoid area: Atrap = ½(a + b) × h
- Multiply by 2 for both end faces: 2 × Atrap
- Find the perimeter of the trapezoid: P = a + b + c + d
- Multiply perimeter by length: P × H = lateral area
- Add: SA = 2 × Atrap + P × H
Worked Example
Given: a = 8 cm, b = 5 cm, c = 4 cm, d = 4 cm, h = 3 cm, H = 12 cm
Step 1: Trapezoid area = ½(8 + 5) × 3 = ½ × 13 × 3 = 19.5 cm²
Step 2: Two ends = 2 × 19.5 = 39 cm²
Step 3: Perimeter = 8 + 5 + 4 + 4 = 21 cm
Step 4: Lateral area = 21 × 12 = 252 cm²
Step 5: SA = 39 + 252 = 291 cm²
Step 1: Trapezoid area = ½(8 + 5) × 3 = ½ × 13 × 3 = 19.5 cm²
Step 2: Two ends = 2 × 19.5 = 39 cm²
Step 3: Perimeter = 8 + 5 + 4 + 4 = 21 cm
Step 4: Lateral area = 21 × 12 = 252 cm²
Step 5: SA = 39 + 252 = 291 cm²
When to Use This Formula
Trapezoidal prisms appear in roof structures, gutters, troughs, and channel cross-sections. Use this formula whenever you need to estimate material for coating, wrapping, or painting a trapezoidal cross-section duct or beam.
Use our Trapezoidal Prism Calculator for instant results. See all formulas, SA vs lateral area, and how to calculate surface area.