CPM Homework Help : GC Problem 10-41

Pentagonal prism, with front edge of pentagon labeled, 6 feet, and height labeled, 12 feet. Calculate the total surface area and volume of the prism at right. Assume that the base is a regular pentagon.

Step 1:

Find the central angle of the pentagon.

$\frac{360°}{5}=72°$

Hint:

The area of the triangle will help you find the area of the pentagon.

A triangle, with bottom side labeled, 6, and dashed line segment, labeled, h, from top vertex, perpendicular to the base, creating 2 right triangles. In the right triangle, on the right, the top angle is labeled, 36 degrees.

Step 2:

$\text{tan}\left(36°\right)=\frac{3}{h}$ , $h=\frac{3}{\text{tan}\left(36°\right)}$ , $h \approx 4.1291 \; \text{ft}$

$\text{Area of the triangle} \approx \left(0.5\right) \left(4.1291\right) \left(6\right) \approx 12.39 \; \text{ft}^2$
$\text{Area of pentagon} \approx \left(5\right) \left(12.39\right) \approx 61.9 \; \text{ft}^2$

Step 3:

Find the area of each of the sides, and sum them to find the surface area.
$\left(12\right)\left(6\right)\left(5\right)=360$
$360 + 2 \left(61.9\right) \approx 483.8 \; \text{ft}^2$

Step 4:

Multiply the area of the pentagon by the height to find the volume.
$\left(61.9\right)\left(12\right)\approx743.3\text{ ft}^3$

Answer:

$\text{Volume} \approx 743.3 \; \text{ ft}^3$
$\text{Surface area} \approx 483.3 \; \text{ft}^2$