Answer to Question #318517 in Geometry for shibs

Question

The base of a prism is a regular pentagon with one side measuring 15 cm. Its

Altitude is 32 cm. Find the following;

1. Lateral surface Area in cm2.
2. The total surface area in cm2.
3. The volume of the prism in cm3.

Answer 1

Let $AB=BC=CD=DE=AE$ then $A$₁$B$₁$=B$₁$C$₁$=C$₁$D$₁$=D$₁$E$₁ $=E$₁$A$₁

Then the area of the upper and lower bases of the prism: $S\scriptsize{base} \normalsize =\frac{AB^2\sqrt{25+10\sqrt{5}}}4=\frac{225\sqrt{25+10\sqrt{5}}}4$,

both bases: $S^*=2S=\frac{225\sqrt{25+10\sqrt{5}}}2$

1.

A side face has the area $S=AB\cdot AA$₁ , all five side faces have the area $S^{**}=5S=5\cdot15\cdot32=2400$ cm²

2.

The total surface area: $S=S^*+S^{**}=\frac{225\sqrt{25+10\sqrt{5}}}2+2400\thickapprox3175,2148$ cm²

3.

The volume of the prism: $V=S\scriptsize{base}\normalsize \cdot AA$₁ $=\frac{225\sqrt{25+10\sqrt{5}}}4\cdot32=1800\sqrt{25+10\sqrt{5}}$ $\thickapprox12387,4373$ cm³