In what ratio are the volumes of a cylinder, a cone and a sphere, if each has the same diameter and the same height?

### Concept & Proportional Volumes To compare the volumes of different geometric shapes with shared dimensions, express all dimensions in terms of a single variable, compute the standard formulas, and evaluate the final ratio. ### Step-by-Step Solution 1. **Identify Common Variables:** Let diameter = $d$. Then radius $r = \frac{d}{2}$ for all shapes. For a sphere, the height is its diameter. Thus, height $h = d = 2r$. Since all shapes have the same height, $h = 2r$ for the cylinder and cone as well. 2. **Express Volumes in terms of $r$:** - **Cylinder:** $V_{cyl} = \pi r^2h = \pi r^2(2r) = 2\pi r^3$ - **Cone:** $V_{cone} = \frac{1}{3}\pi r^2h = \frac{1}{3}\pi r^2(2r) = \frac{2}{3}\pi r^3$ - **Sphere:** $V_{sph} = \frac{4}{3}\pi r^3$ 3. **Calculate the Ratio:** $V_{cyl} : V_{cone} : V_{sph} = 2\pi r^3 : \frac{2}{3}\pi r^3 : \frac{4}{3}\pi r^3$ Divide by $2\pi r^3$: $= 1 : \frac{1}{3} : \frac{2}{3}$ Multiply by 3 to clear the fractions: $= 3 : 1 : 2$ ### Exam Strategy & Shortcut If you recall Archimedes' principle regarding the cylinder and inscribed sphere, a sphere is $\frac{2}{3}$ the volume of its circumscribed cylinder, and a cone is $\frac{1}{3}$ of it. Thus the ratio of Cylinder:Cone:Sphere is $1 : \frac{1}{3} : \frac{2}{3}$, which scales immediately to $3 : 1 : 2$. ### Common Pitfall Misidentifying the "height" of the sphere. The height of a sphere is equal to its diameter ($2r$), not its radius ($r$). ### Final Answer Therefore, the correct answer is **3 : 1 : 2**.