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How to Find the Radius of a Cone?

Published in Geometry 2 mins read

Finding the radius of a cone depends on what information you already have. Here's a breakdown of the different scenarios and formulas you can use:

1. If you know the Volume (V) and Height (h):

This is the most common scenario. You can use the formula for the volume of a cone and rearrange it to solve for the radius (r).

  • Formula: V = (1/3) π r² * h

  • Solving for r:

  1.  Multiply both sides by 3: 3V = π * r² * h
  2.  Divide both sides by π * h:  (3V) / (π * h) = r²
  3.  Take the square root of both sides: r = √((3V) / (π * h))

Example:

Let's say a cone has a volume of 80 cubic inches and a height of 9 inches. Using π ≈ 3.14:

r = √((3 80) / (3.14 9))
r = √(240 / 28.26)
r = √(8.49)
r ≈ 2.91 inches

2. If you know the Slant Height (s) and Height (h):

You can use the Pythagorean theorem since the radius, height, and slant height form a right triangle.

  • Pythagorean Theorem: r² + h² = s²

  • Solving for r:

  1.  Subtract h² from both sides: r² = s² - h²
  2.  Take the square root of both sides: r = √(s² - h²)

3. If you know the Lateral Surface Area (LSA) and Slant Height (s):

  • Formula: LSA = π r s

  • Solving for r:

  1.  Divide both sides by π * s:  r = LSA / (π * s)

4. If you know the Total Surface Area (TSA) and Slant Height (s):

  • Formula: TSA = π r s + π * r²

This equation is a quadratic equation in terms of 'r', and you'll need to use the quadratic formula to solve for r:

TSA = πrs + πr²
πr² + πsr - TSA = 0

Therefore, the radius, r, is:
r = (-πs ± √((πs)² - 4(π)(-TSA))) / (2π)

Important Considerations:

  • Units: Make sure all measurements are in the same units before performing calculations.
  • Approximations: When using π, you can use 3.14 or the π button on your calculator for more accuracy. Your answer will be an approximation if you use an approximation of pi.

In summary, determining the radius of a cone depends on the available information. By using the correct formula and rearranging it appropriately, you can accurately calculate the radius.

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