Frustum of a Cone

What Is a Frustum?

Take a cone and slice off the pointed top with a cut parallel to the base. The piece left at the bottom — a cone with its tip removed — is a frustum of a cone. A bucket, a drinking glass, a lampshade and a flowerpot are everyday frustums.

A frustum has two circular faces: a bigger base of radius $R$ and a smaller top of radius $r$, separated by a vertical height $h$ and a slant side $l$.

The Slant Height

The slant side of a frustum is the hypotenuse of a right triangle whose legs are the height $h$ and the difference of the radii $(R-r)$: $$\boxed{l = \sqrt{h^2 + (R-r)^2}}$$

Key Point: Use $(R-r)$, the difference of the radii — not $R+r$ — inside the slant-height formula. Find $l$ first if the question gives $h$, $R$, $r$.

Surface Areas of a Frustum

• Curved (lateral) surface area: $\text{CSA} = \pi l (R + r)$.
• Total surface area: add both circular faces: $$\text{TSA} = \pi l(R+r) + \pi R^2 + \pi r^2.$$

For an open bucket, there is no top face, so its surface (metal used) $= \pi l(R+r) + \pi r^2$ (curved side $+$ the smaller closed base).

Key Point: CSA uses $(R+r)$; the two flat faces are $\pi R^2$ and $\pi r^2$. Read whether the object is open or closed.

Volume of a Frustum

$$\boxed{V = \dfrac13\pi h\left(R^2 + r^2 + Rr\right)}$$

This neatly reduces to a full cone when $r = 0$ ($V = \tfrac13\pi h R^2$) and to a cylinder when $R = r$ ($V = \pi h R^2$) — a good way to remember it.

[Board Important] Capacity of a bucket $= \dfrac13\pi h(R^2 + r^2 + Rr)$; give the answer in litres if asked ($1\text{ litre} = 1000\text{ cm}^3$).

Solved Examples

Example 1: Slant height and CSA

A frustum has radii $R = 20$ cm and $r = 8$ cm and vertical height $h = 16$ cm. Find its slant height and curved surface area. $\left(\pi=3.14\right)$

Solution:

1. $l = \sqrt{h^2 + (R-r)^2} = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20$ cm.
2. CSA $= \pi l(R+r) = 3.14\times 20\times 28 = 1758.4$ cm$^2$.

Final Answer: $l = 20$ cm; CSA $= 1758.4$ cm$^2$.

Takeaway: $(R-r)=12$ and $h=16$ give the $12,16,20$ triple.

Example 2: Volume of a bucket

A bucket is a frustum with $R = 20$ cm, $r = 8$ cm and height $h = 16$ cm. Find its capacity in litres. $\left(\pi=3.14\right)$

Solution:

1. $V = \dfrac13\pi h(R^2 + r^2 + Rr) = \dfrac13\times 3.14\times 16\times(400 + 64 + 160)$.
2. $= \dfrac13\times 3.14\times 16\times 624 = 10449.92$ cm$^3$.
3. In litres $= \dfrac{10449.92}{1000} \approx 10.45$ L.

Final Answer: $\approx 10.45$ litres.

Takeaway: $V = \tfrac13\pi h(R^2+r^2+Rr)$; divide by 1000 for litres.

Example 3: Metal in an open bucket

Find the area of metal sheet used to make the open bucket of Example 1 ($R=20$, $r=8$, $l=20$ cm). $\left(\pi=3.14\right)$

Solution:

1. Open bucket $=$ curved side $+$ smaller base $= \pi l(R+r) + \pi r^2$.
2. $= 3.14\times 20\times 28 + 3.14\times 64 = 1758.4 + 200.96 = 1959.36$ cm$^2$.

Final Answer: $1959.36$ cm$^2$.

Takeaway: Open bucket has no top face — add only the smaller base.

Example 4: Frustum glass

A glass is a frustum with top radius 3 cm, bottom radius 2 cm and height 6 cm. Find its capacity. $\left(\pi=3.14\right)$

Solution:

1. Here $R = 3$, $r = 2$, $h = 6$.
2. $V = \dfrac13\times 3.14\times 6\times(9 + 4 + 6) = \dfrac13\times 3.14\times 6\times 19 = 119.32$ cm$^3$.

Final Answer: $\approx 119.32$ cm$^3$.

Takeaway: It does not matter which end you call $R$ or $r$ in the volume formula — $R^2+r^2+Rr$ is symmetric.