Answer to Question #138884 in Geometry for cocolex

Question

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

Answer 1

4.The formula of a triangle area by two sides and the angle between them is:

$S=\frac{1}{2}ab\sin\gamma$

since the triangle is equilateral

$a=b=c$ $and$ $\gamma =\frac{\pi}{3}; \,\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\,$

$S=\frac{\sqrt{3}a^2}{4};\,from \,here\, a =\sqrt{\frac{4S}{\sqrt{3}}}$

$a =\sqrt{\frac{4*1560}{\sqrt{3}}}\approx60,02cm(1)$

5.The lateral area  of a right prism can be calculated by multiplying the perimeter of the base by the height of the prism.

Since at the base of the prism is an equilateral triangle, the perimeter of the base:

$P = 3*a$

where a in meters from (1), $a\approx0.6 m$

$h= 5m;$ from the task condition

$S_l=P*h = 3*a*h =3*0.6*5 =9m^2$

6.The volume of a right prism can be calculated by multiplying the aquare of the base by the height of the prism.

$V = S_b*h = 0.156*5= 0.78m^3$

where $S_b = 1560cm^2 =1560/100^2=0.156m^2$

Answer : base edge of the prism 60.02 cm;

lateral area of the prism 9m²;

volume of the prism  0.78m³