Skill: 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.