Surface Area of Prisms and Cylinders Examples

Find the lateral area and the surface area of the solid.

The first thing we should know is that since it's a rectangular prism, there could be 3 different pairs of bases here. To start, we'll stay that the 7 cm × 2 cm side is our base.

SA = L + 2B
SA = Ph + 2B

The perimeter of our base is 7 + 2 + 7 + 2, or 18 cm. The height is 13 cm. Our rectangular bases are equal to their length times their width, or 7 cm × 2 cm = 14 cm².

SA = (18 cm)(13 cm) + 2(14 cm²)
SA = 234 cm² + 28 cm² = 262 cm²

That means the surface area is 262 cm², and the lateral area when the base is 7 cm × 2 cm is 234 cm².

If the bases were the 7 cm × 13 cm sides, our lateral surface area would be different. The perimeter would be 7 + 13 + 7 + 13 = 40 cm, and our height would be 2 cm. Since L = Ph, our lateral area would be (40 cm)(2 cm) = 80 cm².

One last base pair: the 2 cm × 13 cm sides. The perimeter would be 30 cm and the height is 7 cm, so the lateral area is L = Ph = (30 cm)(7 cm) = 210 cm².

Notice that while the lateral area changes depending on the bases we use, the surface area for the prism stays the same. That makes sense, because we don't use a different amount of wrapping paper just because the object is facing a different way. Unless you do, which would be weird.

Find the lateral area and the surface area of the cylinder.

Like always, we'll use our handy dandy surface area formula, SA = L + 2B. In our case, the lateral area equals 2πrh and each base is the area of a circle, πr². That's a lot of pi.

SA = 2πrh + 2(πr²)

Substituting in the values we know, we get:

SA = 2π(1.2 in)(7.3 in) + 2π(1.2 in)²
SA = 2π(8.76 in²) + 2π(1.44 in²)
SA = 55.04 in² + 9.05 in² = 64.09 in²

Ta-da! In one fell swoop, we calculated the surface area (64.09 in²) and the lateral area (55.04 in²). Check and check.

The lateral area of a regular hexagonal prism is 81 ft². If the prism has a height of 3 ft, what is the length of one side of the hexagonal base?

Since the hexagonal base is regular (thanks to Activia yogurt), all its sides are the same length—let's call it x. The lateral area equals the perimeter times the height.

L = Ph

The height is 3 and the lateral area is 81. The perimeter P of the hexagon is 6x, so we can substitute that in, too. Solving for x will give us our hexagonal side length.

81 = (6x)(3)
81 = 18x

Find the lateral and surface areas of the solid.

This is a triangular prism. First thing we gotta know is the perimeter of the triangle. It's special triangle and/or Pythagorean Theorem time.

a² + b² = c²
10² + 10² = c²
200 = c²
c ≈ 14.1 yd


Woohoo. That means we can calculate the perimeter of the triangle.

P = 10 yd + 10 yd + 14.1 yd = 34.1 yd

Now that we know the perimeter and the height, here comes the lateral area.

L = Ph = (34.1 yd)(15 yd)
L = 511.5 yd²

To finish off the entire surface area, we need to add the areas of the two triangular bases. The area of a triangle is ½bh.

SA = L + 2B
SA = 511.5 yd² + 2(½bh)
SA = 511.5 yd² + 2(½)(10 yd)(10 yd) = 611.5 yd²

You did it. Give yourself a pat on the back. Or a Nintendo Wii.