Calculate the surface area of prisms

Surface area of a prism is the total area of its two congruent, parallel bases and its rectangular lateral faces, found from a net or the relationship $SA=2B+Ph$ , where $B$ is the area of a base, $P$ its perimeter, and $h$ the prism’s length or height. The reasoning includes selecting and combining appropriate area formulas for rectangular, triangular, and other polygonal bases, reporting square units, and distinguishing surface area from volume; curved, oblique, and more advanced generalized solids are not included.

Detailed Explanation: Calculate the surface area of prisms

Surface area is the total area of every outside face of a prism. A prism has two congruent bases and rectangular lateral faces.

Use the formula

SA=2B+Ph

where:

$B$ is the area of one base,
$P$ is the perimeter of one base,
$h$ is the length of the prism.

Example

A triangular prism has right-triangular bases with side lengths $3\text{ cm}$ , $4\text{ cm}$ , and $5\text{ cm}$ . The prism is $10\text{ cm}$ long. Find its surface area.

Step 1: Find the area of one triangular base

Use the area formula for a triangle:

B=\frac{1}{2}bh

Using the perpendicular sides of lengths $3$ cm and $4$ cm:

B=\frac{1}{2}(3)(4)=6\text{ cm}^2

Step 2: Find the perimeter of the base

Add the three side lengths:

P=3+4+5=12\text{ cm}

Step 3: Identify the prism’s length

The prism is $10$ cm long, so

h=10\text{ cm}

Step 4: Substitute into the surface area formula

SA=2B+Ph

SA=2(6)+(12)(10)

SA=12+120=132\text{ cm}^2

Therefore, the surface area of the triangular prism is

\boxed{132\text{ cm}^2}

The answer uses square units because surface area measures the amount of two-dimensional space covering the outside of the prism. Do not confuse it with volume, which measures the space inside the prism and uses cubic units.

Learn by doing: Calculate the surface area of prisms

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Surface Area of a Rectangular Prism - Calculate

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