The surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 6 cm each. What is the radius (in cm) of the sphere?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the given values
We are given:
- Height of the cylinder (h) = 6 cm
- Diameter of the cylinder = 6 cm

From the diameter, we can find the radius of the cylinder (r_cylinder):
\[ \text{Radius of the cylinder} = \frac{\text{Diameter}}{2} = \frac{6 \text{ cm}}{2} = 3 \text{ cm} \]

### Step 2: Write the formulas for surface areas
The surface area of a sphere (SA_sphere) is given by:
\[ SA_{\text{sphere}} = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

The curved surface area of a right circular cylinder (CSA_cylinder) is given by:
\[ CSA_{\text{cylinder}} = 2\pi r_{\text{cylinder}} h \]
where \( r_{\text{cylinder}} \) is the radius of the cylinder and \( h \) is the height of the cylinder.

### Step 3: Substitute the values into the formulas
We know that the surface area of the sphere is equal to the curved surface area of the cylinder:
\[ 4\pi r^2 = 2\pi r_{\text{cylinder}} h \]

Substituting the known values:
- \( r_{\text{cylinder}} = 3 \text{ cm} \)
- \( h = 6 \text{ cm} \)

So, we have:
\[ 4\pi r^2 = 2\pi (3)(6) \]

### Step 4: Simplify the equation
First, simplify the right side:
\[ 2\pi (3)(6) = 36\pi \]

Now, we have:
\[ 4\pi r^2 = 36\pi \]

### Step 5: Cancel out \(\pi\) from both sides
Dividing both sides by \(\pi\):
\[ 4r^2 = 36 \]

### Step 6: Solve for \( r^2 \)
Now, divide both sides by 4:
\[ r^2 = \frac{36}{4} = 9 \]

### Step 7: Take the square root to find \( r \)
Taking the square root of both sides:
\[ r = \sqrt{9} = 3 \text{ cm} \]

### Conclusion
The radius of the sphere is \( 3 \text{ cm} \).

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