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Computing Surface Area of Three-dimensional Solids

Filed Under: Aviation Mathematics

The surface area of a three-dimensional solid is the sum of the areas of the faces of the solid. Surface area is a different concept from that of volume. For example, surface area is the amount of sheet metal needed to build a rectangular fuel tank while volume is the amount of fuel that the tank can contain.

Rectangular Solid

The formula for the surface area of a rectangular solid [Figure 1-24] is given as:

Figure 1-24. Rectangular solid.
Figure 1-24. Rectangular solid.

Surface Area =

2 × [(Width × Length) + (Width × Height) + (Length × Height)] = 2 × [(W × L) + (W × H) + (L × H)]
 

Cube

The formula for the surface area of a cube [Figure 1‑25] is given as:

Figure 1-25. Cube.
Figure 1-25. Cube.

Surface Area = 6 × (Side × Side) = 6 × S2

Example: What is the surface area of a cube with a side measure of 8 inches?

Surface Area = 6 × (Side × Side)

= 6 × S2 = 6 × 82 = 6 × 64
= 384 square inches
 

Cylinder

The formula for the surface area of a cylinder [Figure 1-26] is given as:

Figure 1-26. Cylinder.
Figure 1-26. Cylinder.
Surface Area = 2 × π × radius2 + π × diameter × height
= 2 × π × r2 + π × D × H
 

Sphere

The formula for the surface area of a sphere [Figure 1-28] is given as:

Figure 1-28. Sphere.
Figure 1-28. Sphere.
Surface Area = 4 × π × radius2 = 4 × π × r2
 

Cone

The formula for the surface area of a right circular cone [Figure 1-29] is given as:

Figure 1-29. Cone.
Figure 1-29. Cone.
Surface Area = π × radius × [radius + (radius2 + height2)1⁄2]
= π × r × [r + (r2 + H2)1⁄2]
 

Figure 1-30 summarizes the formulas for computing the volume and surface area of three-dimensional solids.

Figure 1-30. Formulas to compute volume and surface area.
Figure 1-30. Formulas to compute volume
and surface area.

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