The surface areas of two similar shapes are in the ratio 4:9. What is the ratio of their volumes?

Question

The surface areas of two similar shapes are in the ratio 4:9. What is the ratio of their volumes?

Answer 1

If they are similar, then all linear measurements are proportional. All areas are approximated as sums of small square areas. Sides of the squares are proportional, too. There areas computed as squares of their sides, so if the areas of two shapes are in ratio 4:9, then their linear measurements must be in ratio √4: √9, or 2:3, to keep the ratio of square areas as given. We can approximate a volume as a sum of very small cubes, filling a shape. A volume of a cube is a cube of its side length. Therefore the resulting ratio of volumes is 2³: 3³, or 8:27.

Answer 2

1. Since the ratio of their two-dimensional property (i.e., surface area) is 4:9, we can take the square root of this to determine the relationship of their one dimension.
2. Sqrt(4/9) = Sqrt(4) / Sqrt(9)  = 2/3
3. Volume is a three dimensional property, so the ratio of the two shapes is (2/3)³ = 8/27.

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Here's another way to arrive at the same answer, using a cube as the shape.

1. The surface area of a cube is just 6* area of one surface.  The reason for multiplying by 6 is that there are 6 surfaces on a cube.  The area of a surface is the length of a side squared.  Putting it altogether, the Surface Area (SA) of a cube = 6* s²
2. If there are two cubes, C1 and C2, that have a SA ratio of 4:9, then 9*SA_C1 = 4*SA_C2  {Eqn 1}
3. Let s1 = length of each side of C1.  And s2 = length of each side of C2.  Then Eqn 1 can be rewritten as:
1. 9*(6s₁²)  = 4*(6s₂²)
2. 54s₁² = 24s₂²
3. 54/24s₁² = s₂²
4. sqrt(54/24s₁²) = s₂ 
5. s₂ = sqrt(54/24) * sqrt(s₁²)
6. s₂ = sqrt(54/24) * s₁ 
7. s2 = (54/24)^1/2 * s₁ {Eqn 2}
4. The volume V of a cube is simply s³
1. V_C1 =  s₁³
2. V_C2 =  s₂³   {Eqn 3}
3. Substitute Eqn 2 into Eqn 3.  V_C2 = (54/24)^1/2 * s₁)³
4. V_C2 = (54/24)^3/2 * s₁³
5. V_C2 = 3.375s₁³
5. Now that the volumes are both expressed as a function of s₁, we can compare their ratio
6. V_C1/V_C2 = s₁³ / 3.375s₁³
7. V_C1/V_C2 = 0.296296
8. If you want to express this as an integer ratio:  8: 27.