Surface area = 2(lw + lh + wh) = 2[(6.6×9.9) + (6.6×16.5) + (9.9×16.5)] - Parker Core Knowledge
Surface Area Formula Explained: How to Calculate the Surface Area Using 2(lw + lh + wh)
Surface Area Formula Explained: How to Calculate the Surface Area Using 2(lw + lh + wh)
Calculating the surface area of a rectangular prism is essential in fields like architecture, engineering, packaging design, and material estimation. The standard formula is:
Surface Area = 2(lw + lh + wh)
or equivalently,
Surface Area = 2[(l × w) + (l × h) + (w × h)]
Understanding the Context
In this article, we’ll break down how to compute the surface area using real dimensions: length (l) = 6.6 m, width (w) = 9.9 m, and height (h) = 16.5 m, with step-by-step explanation of each component.
Understanding the Formula: Why We Use 2(lw + lh + wh)
A rectangular prism has six faces: two of each pair of opposite faces. Each pair corresponds to one of the three dimensions multiplied together:
Image Gallery
Key Insights
- Two faces with area l × w
- Two faces with area l × h
- Two faces with area w × h
To find the total surface area, we compute the sum of all these face areas and multiply by 2 to account for both sides:
Surface Area = 2(lw + lh + wh)
This efficient formula avoids counting each face twice by summing the three unique pairs and doubling the total.
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Step-by-Step Calculation with Actual Values
Let’s apply the formula using the given dimensions:
l = 6.6 m, w = 9.9 m, h = 16.5 m
Step 1: Calculate each pairwise product
-
l × w = 6.6 × 9.9
= 65.34 m²
(Calculation: 6.6 × 10 = 66, subtract 6.6 × 0.1 = 0.66 → 66 – 0.66 = 65.34) -
l × h = 6.6 × 16.5
= 108.90 m²
(Calculation: 6.6 × 16 = 105.6, plus 6.6 × 0.5 = 3.3 → 105.6 + 3.3 = 108.90) -
w × h = 9.9 × 16.5
= 163.35 m²
(Calculation: 9.9 × 16 = 158.4, plus 9.9 × 0.5 = 4.95 → 158.4 + 4.95 = 163.35)
Step 2: Sum the products
lw + lh + wh = 65.34 + 108.90 + 163.35
= 337.59 m²
Step 3: Multiply by 2 to complete the surface area formula
Surface Area = 2 × 337.59 = 675.18 m²
