To find the length of a side of a cube given its total surface area, follow these steps:
1. Understand the relationship between the surface area and the side length of a cube:
- A cube has 6 faces, each of which is a square.
- If the side length of the cube is [tex]\( s \)[/tex] cm, then the area of one face is [tex]\( s^2 \)[/tex] cm².
- Therefore, the total surface area ([tex]\( A_{\text{total}} \)[/tex]) of the cube is obtained by summing the areas of all six faces:
[tex]\[
A_{\text{total}} = 6 \cdot s^2
\][/tex]
2. Set up the equation using the given total surface area:
- We are given that the total surface area of the cube is [tex]\( 3174 \)[/tex] cm². So, we set up the equation:
[tex]\[
3174 = 6 \cdot s^2
\][/tex]
3. Solve for [tex]\( s^2 \)[/tex]:
- To isolate [tex]\( s^2 \)[/tex], divide both sides of the equation by 6:
[tex]\[
s^2 = \frac{3174}{6}
\][/tex]
- Performing the division gives:
[tex]\[
s^2 = 529
\][/tex]
4. Take the square root to find [tex]\( s \)[/tex]:
- To find the side length [tex]\( s \)[/tex], take the square root of both sides of the equation:
[tex]\[
s = \sqrt{529}
\][/tex]
5. Calculate the square root:
- The square root of 529 is:
[tex]\[
s = 23
\][/tex]
Therefore, the length of a side of the cube is [tex]\( 23 \)[/tex] cm.
