A cone and sphere have equal volumes and radii of equal length. If the height of the cone is 
36 centimeters, then what is the length of the radius of each shape in centimeters?


We are given that a cone and a sphere have equal volumes and equal radii, and that the height of the cone is 36 cm. We need to find the length of the radius \( r \) of both shapes.

### Step 1: Write the formulas for the volumes of the cone and the sphere.

- The volume \( V_{\text{cone}} \) of a cone is given by:
  \[
  V_{\text{cone}} = \frac{1}{3} \pi r^2 h
  \]
  where \( r \) is the radius and \( h \) is the height of the cone.
  
- The volume \( V_{\text{sphere}} \) of a sphere is given by:
  \[
  V_{\text{sphere}} = \frac{4}{3} \pi r^3
  \]
  where \( r \) is the radius of the sphere.

### Step 2: Set the volumes equal to each other.

Since the volumes of the cone and the sphere are equal, we can set the two formulas equal to each other:
\[
\frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3
\]
Since \( \pi \) appears on both sides, it cancels out:
\[
\frac{1}{3} r^2 h = \frac{4}{3} r^3
\]
Multiply both sides by 3 to get rid of the denominator:
\[
r^2 h = 4 r^3
\]
Now, divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[
h = 4r
\]

### Step 3: Substitute the height of the cone.

We are told that the height of the cone is 36 cm. Substituting this value for \( h \):
\[
36 = 4r
\]
Now, solve for \( r \):
\[
r = \frac{36}{4} = 9 \, \text{cm}
\]

### Final Answer:
The radius of both the cone and the sphere is 9 centimeters.

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Would you like further details on any step or have any questions? Here are some related questions:

1. How do you find the volume of a truncated cone?
2. What is the surface area formula for a sphere?
3. Can the volume of a cone be greater than that of a sphere if they have the same radius?
4. What is the relationship between the slant height and radius in a cone?
5. How would the radius change if the height of the cone was doubled?

**Tip:** Always cancel out common terms like \( \pi \) early to simplify equations in geometry problems.

A cone and sphere have equal volumes and radii of equal length. If the height of the cone is 36 centimeters, then what is the length of the radius of each shape in centimeters?

Solve the problem of finding the radius of a cone and sphere with equal volumes and radii, given the height of the cone as 36 cm. This solution involves key geometry concepts like the volume formulas for cones and spheres, suitable for students in Grades 9-10.

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