Answer to Question #138135 in Geometry for Nirvana

Question

The base of the right prism is an equilateral triangle. The area of the base
is 1,560 cm2. If the altitude of the prism is 5 meters,
1. What is the base edge of the prism in cm?
2. What is the lateral area of the prism in m2?
3. What is the volume of the prism in m3?

Answer 1

1. What is the base edge of the prism in cm?

Let the sides of the triangle are $a, b, c$ and angles are $\alpha, \beta, \gamma$ . The formula of a triangle area by two sides and the angle between them is:

$S=1/2*a*b*sin\gamma.$

In equilateral triangle sides are equal $a=b=c$ and angles are equal. Each angle has 60^o. So,

$S(base)=1/2*a^2*sin(60^o), \newline a^2=2*S(base)/sin(60^o), \newline a^2=2*1560/0.866=\newline=3602.77(cm^2),\newline a=√3602.77=60.023 (cm).$

Answer: $60.023 (cm).$

2. What is the lateral area of the prism in m2?

Let the attitude of the prism is

$d. d=5 m. \newline a=60.023 cm=0.6 m.$

A prism's lateral area is equal to one of its bases perimeter times its height:

$S(lateral) =P*d,\newline P =3*a=3*0.6=1.8 (m), \newline S(lateral)=1.8m*5m=\newline=9(m^2).\newline Answer: 9(m^2).$

3. What is the volume of the prism in m3?

$V=S(base) *d, \newline V=0.156 m^2*5m=0.78(m^3).\newline Answer: 0.78 (m^3).$