How to Find the Height of a Cone with the Curved Surface Area?

To find the height of a cone given its curved surface area, you'll need to use the curved surface area formula and some geometric relationships. Here's a step-by-step guide:

1. Understand the Formulas:

• Curved Surface Area (CSA) of a cone: CSA = πrl where:
  • r is the radius of the base of the cone
  • l is the slant height of the cone
• Relationship between height (h), radius (r), and slant height (l): l² = r² + h² (Pythagorean theorem)

2. Identify Known Variables:

• You are given the curved surface area (CSA).
• You need to find the height (h). You'll likely also need to know or be able to determine the radius (r) to solve for h. If you aren't given r, you'll need another piece of information to find it (e.g., the diameter, circumference of the base, or the angle at the vertex).

3. Solve for the Slant Height (l):

• Rearrange the curved surface area formula to solve for l: l = CSA / (πr)

4. Solve for the Height (h):

• Substitute the value of l you found in step 3 into the Pythagorean theorem equation: (CSA / (πr))² = r² + h²
• Rearrange the equation to solve for h: h² = (CSA / (πr))² - r²
• Take the square root of both sides to find h: h = √((CSA / (πr))² - r²)

5. Example:

Let's say the curved surface area (CSA) of a cone is 204.2 cm² and the radius (r) is 6 cm.

• Step 3: Find the slant height (l)
  • l = CSA / (πr) = 204.2 / (π * 6) ≈ 10.84 cm
• Step 4: Find the height (h)
  • h = √((CSA / (πr))² - r²) = √((204.2 / (π * 6))² - 6²) ≈ √(10.84² - 6²) ≈ √(117.51 - 36) ≈ √81.51 ≈ 9.03 cm

Therefore, the height of the cone is approximately 9.03 cm.

In Summary:

1. Use the formula CSA = πrl to find the slant height l, provided you know the curved surface area (CSA) and the radius (r).
2. Use the Pythagorean theorem l² = r² + h² to relate the slant height l, radius r, and height h.
3. Substitute the value of l you found in step 1 into the Pythagorean theorem and solve for h.