How do you find the slant height of a cone using surface area and radius?
a = πr(r+s)
a/(πr) - r = s
To find the slant height of a cone using the surface area and radius, you can follow these steps:
1. Start with the formula for the surface area of a cone: A = πr(r + L), where A is the surface area, r is the radius, and L is the slant height.
2. Rearrange the formula to solve for the slant height (L): L = √(A/π - r^2).
3. Plug in the given values for the surface area (A) and radius (r) into the formula.
4. Calculate the expression inside the square root (√), then take the square root of that expression to find the slant height (L).
Let's work through an example to see how this is done:
Example:
Given a surface area of 100 square units and a radius of 5 units, find the slant height.
1. Start with the formula for the surface area of a cone: A = πr(r + L).
2. Rearrange the formula to solve for the slant height (L): L = √(A/π - r^2).
3. Plug in the values A = 100 and r = 5 into the formula: L = √(100/π - 5^2).
4. Calculate the expression inside the square root: L = √(100/π - 25).
5. Simplify the expression inside the square root: L = √(100/π - 25) ≈ √(15.92).
6. Take the square root of the expression: L ≈ √15.92 ≈ 3.99.
Therefore, the slant height of the cone is approximately 3.99 units.
To find the slant height of a cone using surface area and radius, you can follow these steps:
1. Determine the surface area of the cone using the formula: SA = πr(r + l), where SA is the surface area, r is the radius, and l is the slant height.
2. Rearrange the formula to solve for the slant height (l): l = √(SA/(πr)) - r.
3. Substitute the given radius and surface area values into the formula.
4. Calculate the slant height (l) using the provided values.
By following these steps, you can find the slant height of a cone using the surface area and radius.