[highlight] [b]Frustums[/b] -Frustums consist of 1 large cone, minus a smaller one -The smaller cone's radius is proportional to the large cone's radius -Therefore the smaller cone's height also follows the same ratio and vice versa -To find the volume of the [b]frustum[/b], calculate the volume of the large cone and find its difference with the small cone[br] [/highlight] As you will be needing to find the volume of a cone, you need the following equation: [code] ([pi]r[squared]h) [divide] 3 [r = radius; h = height] [/code] [h3]Example 1[/h3] Find the volume of [b]A[/b] [b](to 1 decimal place)[/b], when its height is [b]32cm[/b] and its diameter is [b]20cm[/b]. [img src='https://res.cloudinary.com/deylrqt2d/image/upload/v1485040923/Shape-1_yliu76.jpg'] [i]First we need to find the proportion of the smaller [b]blue cone[/b] to the [b]entire cone[/b].[/i] [br] To do this, first find the total height: [code]In this case the total height is: [br] 32cm + 8cm = [b]40cm[/b][/code] ([u]To reduce complexity lets label the [i]smaller blue cone[/i] [b]B[/b][/u]).[br] [highlight] Try to simplify both sides to either one or less. [table] [tr] [td]Full cone height[/td][td]:[/td][td]Smaller cone height[/td] [/tr] [tr] [td]40[/td][td]:[/td][td]8[/td] [/tr] [tr] [td]4[/td][td]:[/td][td]0.8[/td] [/tr] [tr] [td]1[/td][td]:[/td][td]0.2[/td] [/tr] [/table] [/highlight] Now we know that the proportion between the heights of the cones is [b]1 : 0.2[/b], we can tell work out the radius of cone [b]B[/b] as well.[br] [u](Though first we'll need to find the radius of the cone [b]A[/b], which would be [b]10cm[/b] as half the [b]diameter[/b], 20cm, is 10cm)[/u][br] Now all we need to do, is multiply to use the proportion we found, to find the radius of [b]B[/b]. [highlight] The proportion that we earlier got means that both the height and radius of [b]B[/b] is [b]0.2[/b] times as big as that of [b]A[/b], or the full cone.[br] Therefore, just how we get the height of [b]B[/b] by multiplying [b]A[/b]'s height by 0.2, we can do the same for the radius. [code] 10 [multiply] 0.2 = 2 [10 is the radius of [b]A[/b]] [/code] [/highlight] So the radius of [b]B[/b] is 2.[br][br] Now we can move on to finding the volume.[br] First, get the volume of the full cone (so both [b]A[/b] and [b]B[/b]'s combined volume): [code] [table] [tr] [td] Volume [b]Full Cone[/b][/td][td][space]=[space][/td][td style='text-align:left']([pi]r[squared]h) [divide] 3[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']([pi] [multiply] 10[squared] [multiply] 40) [divide] 3[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left'](4000[pi]) [divide] 3[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']1333.33.... [multiply] [pi] [/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']4188.790205cm[cubed][/td] [/tr] [/table] (I've left it to [b]7 significant figures[/b], just so I don't lose too much accuracy) [/code] [br] Now we get the volume of [b]B[/b] (the smaller cone): [code] [table] [tr] [td] Volume of [b]B[/b][/td][td][space]=[space][/td][td style='text-align:left']([pi]r[squared]h) [divide] 3[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left'] ([pi] [multiply] 2[squared] [multiply] 8) [divide] 3[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']32[pi] [divide] 3[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']10.666...... [multiply] [pi] [/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']33.5103216cm[cubed][/td] [/tr] [/table] (I've also left it to [b]7 significant figures[/b] here, just so I, again, don't lose too much accuracy) [/code] [br] Now all that's left is to get the volume of [b]A[/b]: [code] [table] [tr] [td] Volume of [b]A[/b][/td][td][space]=[space][/td][td style='text-align:left']([b]Volume of full cone[/b]) - ([b]Volume of B[/b])[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']4188.790205 - 33.5103216[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']4155.2798834[/td] [/tr] [tr] [td][/td][td][space]=[space][/td][td style='text-align:left']4155.3cm[cubed] (to 1 decimal place)[/td] [/tr] [/table] [/code] [highlight] [i]Therefore the volume of [b]A[/b] is [b]4155.3cm[cubed] (to 1 decimal place)[/b][/i] [/highlight]