Surface Area Formula For A Hemisphere With 6 Inch Radius
The question asks us to determine the correct formula for calculating the surface area of a hemisphere with a radius of 6 inches. To solve this, we need to understand the components of a hemisphere's surface area and how they relate to the given radius. This article will walk you through the concepts and calculations necessary to arrive at the correct answer.
Understanding Surface Area of a Hemisphere
A hemisphere is essentially half of a sphere. Therefore, its surface area consists of two parts: the curved surface and the circular base. Let's delve into each of these components to fully grasp how the total surface area is derived.
Curved Surface Area
The curved surface area of a hemisphere is exactly half the surface area of a full sphere. The surface area of a sphere is given by the formula:
S = 4πr²
Where:
- S is the surface area,
- π (pi) is a mathematical constant approximately equal to 3.14159,
- r is the radius of the sphere.
Thus, the curved surface area of a hemisphere is half of this, which is:
Curved Surface Area = (1/2) * 4πr² = 2πr²
In our case, the radius (r) is 6 inches. Plugging this into the formula, we get:
Curved Surface Area = 2π(6)² = 2π(36) = 72π square inches
This part of the surface area accounts for the rounded exterior of the hemisphere. It's crucial to remember that this is only one part of the total surface area.
Circular Base Area
The base of a hemisphere is a circle. The area of a circle is given by the formula:
Area = πr²
Where:
- Area is the area of the circle,
- π (pi) is the same mathematical constant as before,
- r is the radius of the circle.
For our hemisphere with a radius of 6 inches, the area of the circular base is:
Base Area = π(6)² = π(36) = 36π square inches
This circular base forms the flat, bottom surface of the hemisphere. Including this area is essential for calculating the total surface area.
Total Surface Area
To find the total surface area of the hemisphere, we add the curved surface area and the circular base area:
Total Surface Area = Curved Surface Area + Base Area
Total Surface Area = 2πr² + πr²
Combining these, we get:
Total Surface Area = 3πr²
This formula represents the sum of the hemisphere's curved surface and its circular base, giving us the total area that would be covered if we were to, say, paint the entire outer surface of the hemisphere.
Applying the Formula to Our Problem
Now that we have the formula for the total surface area of a hemisphere (S = 3πr²), we can apply it to our specific problem where the radius (r) is 6 inches.
Substituting r = 6 into the formula, we get:
S = 3π(6)²
S = 3Ï€(36)
S = 108Ï€ square inches
This calculation gives us the total surface area of the hemisphere. Now, let's compare this result with the given options to identify the correct formula.
Evaluating the Given Options
We are provided with four options, each representing a different formula for the surface area. Our task is to identify which one matches the correct formula we derived, S = 3π(6)².
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Option A: S = 3π(12)²
This option uses a radius of 12 inches, which is twice the actual radius given in the problem (6 inches). Therefore, this option is incorrect.
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Option B: S = 2π(12)²
Similar to option A, this option also uses a radius of 12 inches and has an incorrect coefficient for the formula. It does not represent the total surface area of a hemisphere. Thus, this option is also incorrect.
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Option C: S = 2π(6)²
This option uses the correct radius of 6 inches but has a coefficient of 2Ï€, which only accounts for the curved surface area of the hemisphere. It omits the area of the circular base, making it an incomplete representation of the total surface area. Hence, this option is incorrect.
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Option D: S = 3π(6)²
This option perfectly matches the formula we derived for the total surface area of a hemisphere with a radius of 6 inches. It includes both the curved surface area and the circular base area. Therefore, this is the correct option.
Conclusion
After a thorough analysis of the surface area components of a hemisphere and a careful evaluation of the given options, we can confidently conclude that Option D: S = 3π(6)² is the correct formula for finding the surface area of a hemisphere with a radius of 6 inches. This formula accurately accounts for both the curved surface and the circular base, providing a comprehensive measure of the hemisphere's total surface area.
Understanding the geometrical properties and formulas associated with shapes like hemispheres is crucial in various fields, including mathematics, physics, and engineering. This detailed explanation should provide a clear understanding of how to approach such problems, ensuring accuracy and confidence in your calculations.
By breaking down the problem into smaller, manageable parts and systematically applying the relevant formulas, we can effectively solve complex geometrical questions. Remember, the key is to understand the underlying concepts and apply them logically to the given scenario.
