Question

Project: Find the relationship among the volumes of a right circular
cone, a hemisphere and a right circular cylinder of equal radii and
equal heights.

Answer 1

The relationship among the volumes of a right circular cone, a hemisphere and a right circular cylinder of equal radii and equal heights is 1:2:3.

Step-by-step explanation:

Step 1:

Since the right circular cone, hemisphere & right circular cylinder have equal radii and equal heights, so,  

Let the radii “r” and heights “h” of all three are denoted as “x” units (as shown in the figure below).

Step 2:

Now, we have

The volume of the right circular cone = $\frac{1}{3}$πr²h = $\frac{1}{3}$πx³ …. (i)

The volume of the hemisphere = $\frac{2}{3}$πr³ = $\frac{2}{3}$πx³ ….. (ii)

And,

The volume of the right circular cylinder = πr²h = πx³ …. (iii)

Step 3:

Thus, from eq. (i), (ii) & (iii), we get

The relationship between the volumes are as follow:

[Volume of the right circular cone] : [Volume of the hemisphere] : [Volume of the right circular cylinder]  

⇔ $\frac{1}{3}$πx³ : $\frac{2}{3}$πx³ : πx³

cancelling all the similar terms

⇔ $\frac{1}{3}$ : $\frac{2}{3}$ : 1

multiplying by 3 throughout

⇔ 1 : 2 : 3