Question

Two cubes have their volumes in the ratio 1:27. Find ratio of surface area.

Answer 1

Let their edges be a and b.
Then,
a³/b³ = 1/27
⇒ (a/b)³ = (1/3)³
⇒ a/b = 1/3 -------------------- (1)
Therefore, Ratio of the surface area,
⇒ 6a²/6b²
⇒ a²/b²
⇒ (a/b)²
⇒(1/3)² [From (1)]
⇒1/9
⇒i.e. 1 : 9
Ratio of the surface area = 1 : 9

Answer 2

Given - Ratio of the volume of cubes

Find - Ratio of surface area

Solution - The ratio of the surface area of two cubes is 1:9.

The volume of cubes is given by the ratio = a³, where a is one side of the cube.

Thus, the volume of cube1:volume of cube2 = a1³:a2³

$\frac{a1}{a2} = \sqrt[3]{ \frac{1}{27} }$

$\frac{a1}{a2} = \frac{1}{3}$

Now, the surface area of the cube is given by the formula = 6a².

$\frac{sa1}{sa2} = \frac{6 {a1}^{2} }{6 {a2}^{2} }$

$\frac{sa1}{sa2} = (\frac{1}{3} ) ^{2}$

$\frac{sa1}{sa2} = \frac{1}{9}$

Therefore, the ratio of the surface area of cubes is 1:9.