Question

the base of a right prism is a rhombus whose diagonals are 6 in and 8. If the altitude of the prism is 12, what is the total surface area of the prism?

Answer 1

The total surface area of the prism is
`SA=288\ in^(2)`

Explanation:

we know that

The two diagonals of a rhombus are perpendicular and bisect each other

All sides are congruent

The surface area of a prism is equal to

`SA=2B+Ph`

where

B is the area of the base of prism

P is the perimeter of the base of prism

h is the height of the prism

step 1

Find the length side of the rhombus

Applying Pythagoras Theorem

`c^(2)=a^(2)+b^(2)`

we have

c is the length side of the rhombus

a and b are the semi diagonals of the rhombus

`a=8/2=4\ in`

`b=6/2=3\ in`

substitute

`c^(2)=4^(2)+3^(2)`

`c^(2)=25`

`c=5\ in`

step 2

Find the perimeter of the base P

The perimeter of the rhombus is equal to

`P=4c`

`P=4(5)=20\ in`

step 3

Find the area of the base B

The area of the rhombus is

`B=(1)/(2)[D1*D2]`

D1 and D2 are the diagonals of the rhombus

substitute

`B=(1)/(2)[8*6]=24\ in^(2)`

step 3

Find the surface area of the prism

`SA=2B+Ph`

substitute the values

`SA=2(24)+(20)(12)`

`SA=288\ in^(2)`