UPSKILL MATH PLUS

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### Theory:

Frustum of a cone:
If a smaller end of the cone is sliced by a plane parallel to its base, the portion of a solid between this plane and the base is known as the frustum of a cone.
Volume of a frustum of a cone:

Let $$R$$ and $$r$$ be the radii of base $$(R > r)$$, $$h$$ be the height, and $$l$$ be the slant height of the frustum of a cone.

We need to find the height of the smaller cone $$ADE$$.

Consider $$\Delta ABC$$ and $$\Delta ADE$$.

$$\angle BAC = \angle DAE$$ [common angle]

$$\angle ABC = \angle ADE$$ [Both $$90^\circ$$]

Therefore, $$\Delta ABC \sim \Delta ADE$$ [by AA similarity].

The corresponding sides of similar triangles are proportional.

$$\frac{AB}{AD} = \frac{BC}{DE}$$

$$\frac{H - h}{H} = \frac{r}{R}$$

$$HR - hR = Hr$$

$$HR - Hr = hR$$

$$H(R - r) = hR$$

$$H = \frac{hR}{(R - r)}$$ - - - - - - (I)

Volume of frustum $$=$$ Volume of big cone $$-$$ Volume of small cone

$$=$$ Volume of $$ADE$$ cone $$-$$ Volume of $$ABC$$ cone

$$=$$ $$\frac{1}{3} \pi R^2H$$ $$-$$ $$\frac{1}{3} \pi r^2(H - h)$$

$$=$$ $$\frac{1}{3} \pi R^2H$$ $$-$$ $$\frac{1}{3} \pi r^2H$$ $$+$$ $$\frac{1}{3} \pi r^2h$$

$$=$$ $$\frac{1}{3} \pi H (R^2 - r^2)$$ $$+$$ $$\frac{1}{3} \pi r^2h$$

$$=$$ $$\frac{1}{3} \pi \times \frac{hR}{(R - r)} (R^2 - r^2)$$ $$+$$ $$\frac{1}{3} \pi r^2h$$ [using equation (I)]

$$=$$ $$\frac{1}{3} \pi \times \frac{hR}{(R - r)} (R - r) (R + r)$$ $$+$$ $$\frac{1}{3} \pi r^2h$$

$$=$$ $$\frac{1}{3} \pi hR(R + r)$$ $$+$$ $$\frac{1}{3} \pi r^2h$$

$$=$$ $$\frac{1}{3} \pi h[R(R + r) + r^2]$$

$$=$$ $$\frac{1}{3} \pi h[R^2 + Rr + r^2]$$
Volume of the frustum of a cone $$=$$ $$\frac{1}{3} \pi h[R^2 + Rr + r^2]$$ cu. units