Anybody good at math? particularly geometry and trig!! ?

Question

a right prism has a height of 34cm. Its base is a regular decagon with a side length of 16.8 cm. Find the surface area and volume of the prism.

i'm for some reason having problems with this question.

and i think it's cause I don't have a diagram!

help me please!!??

Answer 1

You don't need a complex diagram. Just draw an isosceles triangle representing a tenth of the base, and a rectangle representing each face.

Let:

h be the height,

s be the side length of the base,

A be the surface area of the base,

V be the volume.

Dividing the base into 10 congruent triangles meeting at the centre, the central angle of each is 36 deg.

If x is the perpendicular distance from the centre to the base edge, the area a of one triangle is:

a = sx / 2 ...(1)

Dividing the isosceles triangle into two right triangles:

x / (s / 2) = cot(18)

2x / s = cot(18)

x = s cot(18) / 2.

Substituting for x in (1):

a = s^2 cot(18) / 4

The total surface area A of the prism is twice the area of the base (that's 20 times each isosceles triangle) plus 10 times the area of one vertical face:

A = 20a + 10sh

= 5s^2 cot(18) + 10sh

= 5 * 16.8^2 cot(18) + 10 * 16.8 * 34

= 10055 cm^2.

The volume V is the area of the base multiplied by the height:

V = 10ah

= 5s^2 h cot(18) / 2

= 5 * 16.8^2 * 34 cot(18) / 2

= 73835 cm^3.