I am reading math book that has following statement:
- Certain sphere and cube have same volume
- Area of the cube can be calculated using formula
$$ A_{cube} = 6 * \Biggl(\sqrt[3]{\frac{4π}{3}*\biggl(\sqrt[2]\frac{A}{4π}\biggl)^3}\Biggl) $$
where $A$ is area of the sphere.
- $ A = 3.847m^2 $
- $ A_{cube} $ should be $ 4.773m^2$
When I try calculate result of the formula I get different result: $ A_{cube} = 5.351m^2 $. I get same result with my calculator and when using Wolfram Alpha.
I can get to expected result diffently:
- Calculate volume of the sphere from area of the sphere
- Since the cube must have same volume as the sphere I know volume of the cube and can use it to calculate its area
- This leads to expected result of $ 4.773m^2$
I found that ratio between areas of cube and sphere should be close to $ 1.24 $. $ 1.24 * 3.847m^2 \approx 4.77m^2 $ which also is consistent with expected result.
So why do I get different result using the formula?
That equation can be abbreviated as: $6*V_{sphere}^{1/3}$
However, $V_{sphere}^{1/3}$ is the length of each edge on the cube-- you need the area of each face. The equation should be $6*V_{sphere}^{2/3}$. This equation will get you the expected answer. The unabbreviated version of the true equation is:
$$ A_{cube} = 6 * \Biggl(\sqrt[3]{\frac{4π}{3}*\biggl(\sqrt[2]\frac{A}{4π}\biggl)^3}\Biggl)^2 $$