# Thread: Similar shapes (Volume and surface area)

1. ## Similar shapes (Volume and surface area)

Oil is stored in either small drums or large drums. The shapes of the drums are mathematically similar.

A small drum has a volume of 0.006m3 and a surface area of 0.2m2

The height of a large drum is 3 times the height of a small drum.

1) Calculate the volume of a large drum?

How do I go about doing this as I know it's not 3 x 0.006

2) The cost of making a drum is £1.20 for each m2 of surface area. A company wants to store 3240m3 of oil in large drums. Calculate the cost of making enough large drums to store this oil?

Any help would be much appreciated.

2. Let h, s, v be the height, surface area and volume, respectively, of the small drum, and let H, S, V be the height, surface area and volume of the large drum. Since the drums are similar, $S/s = (H/h)^2$ and $V/v=(H/h)^3$. Since you know H/h, s and v, you can find S and V.

3. Why do we have to square H/h?

4. 1) V = 0.006 x 27 = 0.162m3

2) 3240 / 0.162 = 20,000 large drums

so 20,000 x 1.2 = £24,000

5. That's a property of similar figures. You can prove it in some specific cases. For example, suppose drum are cylinders, the diameter of the small drum is r and that of the large drum is R. By the definition of similarity, R / r = H / h. Let's call this ratio k. Then $S = 2\pi R^2 + 2\pi R H$ (top and bottom circles plus the side). So, $S = 2\pi(kr)^2+2\pi(kr)(kh)=k^2(2\pi r^2+2\pi rh)=k^2s$. The principle is true for arbitrary similar figures, though. One probably needs calculus to prove it in general.

6. so 20,000 x 1.2 = £24,000
You need to multiply £1.2 by the total number of square meters, not by the number of drums. For this, you need to find the surface area S of the large drum and multiply it by 20,000.