What is the formula for the total surface area of a cone?
A. πr(h + r)
B. 2πrh
C. πr2 + 2πrh
D. πr2
Explanation:
The total surface area of a cone consists of two parts: the circular base and the lateral surface. The formula for the total surface area of a cone is:
Total Surface Area = πr² + Lateral Surface Area
where r is the radius of the circular base, and h is the height of the cone.
The lateral surface area of a cone is given by the formula:
Lateral Surface Area = πrℓ
where ℓ is the slant height of the cone.
To calculate the slant height, we can use the Pythagorean theorem:
ℓ² = r² + h²
Therefore, ℓ = √(r² + h²)
Substituting this value for ℓ into the formula for the lateral surface area, we get:
Lateral Surface Area = πr√(r² + h²)
Combining this with the formula for the total surface area, we get:
Total Surface Area = πr² + πr√(r² + h²)
Simplifying this expression further, we get:
Total Surface Area = πr² + 2πrh
Therefore, the correct answer is option C (πr² + 2πrh).