Final answer:
The surface area of a cone with a radius of 5 units and a height of 10 units is approximately 254.47 square units, calculated by finding the slant height using the Pythagorean theorem and summing the base and lateral surface areas.
Step-by-step explanation:
Calculating the Surface Area of a Cone
To calculate the surface area of a cone, you need to know two things: the radius of the base (r) and the slant height (l). The formula for the surface area of a cone is A = πr(r + l). Since we have a radius of 5 units and a height of 10 units, we need to find the slant height using the Pythagorean theorem: l = √(r² + h²) which gives us l = √(5² + 10²) = √(25 + 100) = √125. Therefore, the slant height is approximately 11.18 units.
Now, we can use the radius and slant height to calculate the surface area:
- First, find the base area: Base area (Abase) = πr² = 3.14159 × 5² ≈ 78.54 units².
- Next, calculate the lateral surface area: Lateral surface area (Alateral) = πrl = 3.14159 × 5 × 11.18 ≈ 175.93 units².
- Add both areas together to get the total surface area: Total surface area (Atotal) = Abase + Alateral ≈ 78.54 units² + 175.93 units² ≈ 254.47 units².