Finding missing radius in cone volume problems
Finding the missing radius in cone volume problems is a Grade 7 algebra and geometry skill in Big Ideas Math Advanced 2, Chapter 8: Volume and Similar Solids. Given a cone volume and height, solve for radius by rearranging V equals one-third pi r squared h: multiply both sides by 3, divide by pi h to isolate r squared, then take the positive square root. For example, a cone with volume 24 pi cubic inches and height 9 inches has radius equal to the square root of 8.
Key Concepts
When the volume and height of a cone are known, you can find the radius by rearranging the cone volume formula $V = \frac{1}{3}\pi r^2 h$. Isolate $r^2$ by multiplying both sides by 3 and dividing by $\pi h$: $$3V = \pi r^2 h$$ $$\frac{3V}{\pi h} = r^2$$ $$r = \sqrt{\frac{3V}{\pi h}}$$ Since radius is a physical dimension, we use only the positive square root.
Common Questions
How do you find the radius of a cone given its volume and height?
Rearrange the cone volume formula: start with V equals one-third pi r squared h, multiply both sides by 3, divide by pi h to get r squared equals 3V divided by (pi h), then take the square root.
What is the formula for cone radius given volume and height?
r equals the square root of (3V divided by pi h). Substitute the known volume and height, then simplify the fraction before taking the square root.
Why do you use only the positive square root for cone radius?
Radius is a physical measurement representing distance, which must be positive. Even though squared equations have both positive and negative roots, only the positive value makes sense for a physical dimension.
What textbook covers finding cone radius from volume in Grade 7?
Big Ideas Math Advanced 2, Chapter 8: Volume and Similar Solids covers finding missing cone dimensions including radius from volume and height.