Home > GC > Chapter 10 > Lesson 10.1.4 > Problem10-41

10-41.

Calculate the total surface area and volume of the prism below. Assume that the base is a regular pentagon.

Find the central angle of the pentagon.

$\frac{360^{\circ}}{5}=72^{\circ}$

$\text{tan}(36^{\circ})=\frac{3}{h} \ \ h=\frac{3}{\text{tan}(36^{\circ})} \ \ h \approx 4.1291 \text{ft}$

Area of the triangle $= (0.5)(4.1291)(6) ≈ 12.39$ ft$^2$
Area of pentagon $= (5)(12.39) ≈ 61.9$ ft$^2$

Find the area of each of the sides, and sum them to find the surface area.
$(12)(6)(5) = 360$
$360+2(61.9)\approx483.8$ ft$^2$

Multiply the area of the pentagon by the height to find the volume.
$(61.9)(12) ≈ 743.3$ ft$^3$

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

Volume $≈ 743.3$ ft$^3$
Surface area $\approx483.3$ ft$^2$