Question

Both figures are hexagonal prisms.
Figure #1: The side lengths of the hexagonal base are all 8 cm. The apothem of the hexagon is 4√3 cm (4 square root 3). The height of the prism is 20 cm.
Figure #2: The side lengths of the second hexagonal base are 32 cm. The apothem of this hexagon is 16√3 (16 square root 3). The height of this prism is 80 cm.

I need the similarity ratio, the ratio of the surface areas and the volumes.

Answer 1

OK, so all you have to do is multiply the apothem by 6 to get the area of the hexagon. So that's 24 √3. Then times 8 and you get 192. For the second one, multiply the apothem times 6 again and you have 96√3. Then multiply that by 32 and you get 3072.

So your surface area ratio is 8:96 or 1:12 and your volume ratio is 32:3072 which simplifies to 1:96.

Answer 2

Oh, I see. So the volume ratio is 8 times the surface area ratio. Is that right?

Answer 3

Exactly!

Answer 4

Thank you so much :)

Answer 5

The theorem at work here is the following:

If the scale factor of two similar solids is a:b, then:
(1) the ratio of corresponding perimeters is a:b
(2) the ratio of the base areas, the lateral areas, and the total areas is a^2 : b^2
(3) the ratio of the volumes is a^3 : b^3

Answer 6

The scale factor (similarity ratio) is 1:4 or 1/4.

This comes from the ratio of corresponding parts 4 Sqr(3) / 16 Sqr(3) which equals 4/16 = 1/4.

Also,
it is confirmed by the ratio of corresponding parts with measures 20/80 = 1/4.

Answer 7

Returning to part two of the theorem to answer > I need the ratio of the surface areas

(2) the ratio of the base areas, the lateral areas, and the total areas is a^2 : b^2

The ratio of the surface areas of the two solids is (1/4) ^ 2 = 1/16 or 1:16.

It is NOT 1:12. @bakergirl

Answer 8

Returning to part three of the theorem to answer > I need the ratio of the volumes.

The ratio of the volumes is (1/4)^3 = 1/64 or 1:64

It is NOT 1:96. @Mathmagic18

Answer 9

Directrix - can you please contact me privately?