Find the total surface area of a quadrant of a sphere of radius 'r'

Answer

To find the total surface area of a quadrant of a sphere with radius 'r', we can follow these steps:

### Step 1: Understand the Problem
A quadrant of a sphere is one-fourth of the entire sphere. We need to find the total surface area, which includes both the curved surface area and the flat circular areas.

### Step 2: Calculate the Total Surface Area of the Sphere
The formula for the total surface area (TSA) of a sphere is given by:
\[ \text{TSA of Sphere} = 4\pi r^2 \]

### Step 3: Calculate the Curved Surface Area of the Quadrant
Since we are dealing with a quadrant (one-fourth of the sphere), the curved surface area (CSA) of the quadrant will be:
\[ \text{CSA of Quadrant} = \frac{1}{4} \times \text{TSA of Sphere} = \frac{1}{4} \times 4\pi r^2 = \pi r^2 \]

### Step 4: Calculate the Area of the Flat Circular Faces
A quadrant has two flat circular faces (semicircles). The area of one semicircle with radius 'r' is:
\[ \text{Area of one semicircle} = \frac{1}{2} \pi r^2 \]
Since there are two semicircles, the total area of the flat circular faces is:
\[ \text{Total Area of Semicircles} = 2 \times \frac{1}{2} \pi r^2 = \pi r^2 \]

### Step 5: Combine the Areas
Now, we add the curved surface area and the area of the flat circular faces to get the total surface area of the quadrant:
\[ \text{Total Surface Area of Quadrant} = \text{CSA of Quadrant} + \text{Total Area of Semicircles} \]
\[ \text{Total Surface Area of Quadrant} = \pi r^2 + \pi r^2 = 2\pi r^2 \]

### Final Answer
Thus, the total surface area of the quadrant of the sphere is:
\[ \boxed{2\pi r^2} \]