What is the total surface area of the hemisphere?

• (A) 4πr²
• (B) 3πr²
• (C) ⅔πr²
• (D) ⅔4πr²

To find the total surface area of a hemisphere, we need to consider both the curved surface area and the area of the flat circular base. Let's break this down step by step.

Understanding the Components

A hemisphere is half of a sphere. The formulas we need are:

• Curved Surface Area of a Hemisphere: This is given by the formula 2πr², where r is the radius of the hemisphere.
• Area of the Base: The base of the hemisphere is a circle, and its area is calculated using the formula πr².

Calculating Total Surface Area

To find the total surface area of the hemisphere, we add the curved surface area to the area of the base:

Total Surface Area = Curved Surface Area + Base Area

Substituting the formulas we have:

Total Surface Area = 2πr² + πr²

Simplifying the Expression

Now, we can combine the two terms:

Total Surface Area = 2πr² + πr² = 3πr²

Final Answer

Thus, the total surface area of the hemisphere is 3πr². Therefore, the correct answer is (B) 3πr².

Visualizing the Concept

Imagine a half of a basketball. The outer surface you can touch represents the curved surface area, while the flat part where it sits on a table represents the base. When calculating the total surface area, you account for both parts to understand the complete area that is exposed.

In summary, when dealing with hemispheres, remember to consider both the curved surface and the flat base to arrive at the total surface area. This approach can be applied to various geometric shapes, ensuring a comprehensive understanding of their properties.