Question 1181731
Here's how to derive the formula for the volume of the inscribed sphere in terms of *r* and *h*:

**1. Diagram and Key Relationships:**

Draw a cross-section of the cone and sphere. You'll see a circle (representing the sphere) inscribed in a triangle (representing the cone).

*   Let *R* be the radius of the inscribed sphere.
*   The radius of the cone's base is *r*.
*   The height (altitude) of the cone is *h*.
*   The slant height of the cone (the hypotenuse of the triangle) is *s* = sqrt(r² + h²).

**2. Similar Triangles:**

There are two similar right triangles in the cross-section:

*   The large triangle representing the cone, with sides *r*, *h*, and *s*.
*   A smaller triangle formed by the radius of the sphere (*R*), the difference between the cone's height and the sphere's radius (*h - R*), and a portion of the slant height.

The ratio of corresponding sides in similar triangles is equal:

R / r = (h - R) / s

**3. Solve for R:**

R / r = (h - R) / sqrt(r² + h²)
R * sqrt(r² + h²) = r(h - R)
R * sqrt(r² + h²) = rh - rR
R * sqrt(r² + h²) + rR = rh
R(sqrt(r² + h²) + r) = rh
R = rh / (sqrt(r² + h²) + r)

**4. Rationalize the Denominator (Optional but often preferred):**

Multiply the numerator and denominator by the conjugate of the denominator:

R = rh(sqrt(r² + h²) - r) / ((sqrt(r² + h²) + r)(sqrt(r² + h²) - r))
R = rh(sqrt(r² + h²) - r) / (r² + h² - r²)
R = rh(sqrt(r² + h²) - r) / h
R = r(sqrt(r² + h²) - r)

**5. Volume of the Sphere:**

The volume *V* of a sphere is given by:

V = (4/3)πR³

Substitute the expression for *R* we derived:

V = (4/3)π[r(sqrt(r² + h²) - r)]³

Therefore, the volume of the inscribed sphere in terms of *r* and *h* is:

V = (4/3)πr³(sqrt(r² + h²) - r)³