Calculate the surface area of prisms

Surface area of a prism is the total area of its two congruent, parallel bases and its lateral faces, expressed in square units. The relationship $SA=2B+Ph$ for a right prism connects the base area $B$ , base perimeter $P$ , and prism height $h$ , and can be interpreted through nets or decomposed faces; it distinguishes surface area from volume and avoids counting shared edges as exposed faces. Oblique prisms and more advanced generalized treatments are outside this scope.

Detailed Explanation: Calculate the surface area of prisms

To find the surface area of a right prism, add the areas of all its exposed faces:

SA=2B+Ph

where:

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

The $2B$ represents the two congruent bases, and $Ph$ represents the total area of the lateral faces.

Example

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

Step 1: Find the area of one triangular base.

The triangle is a right triangle, so use:

B=\frac{1}{2}bh

Using the legs $3$ cm and $4$ cm:

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

Step 2: Find the perimeter of the base.

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

Step 3: Identify the prism height.

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}

Remember that surface area is measured in square units because it measures the total area of the outside faces.

Learn by doing: Calculate the surface area of prisms

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Surface Area of a Rectangular Prism - Calculate

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