A frustum has radii 5cm and 10cm and height 12cm — find volume and surface area

Question

A frustum of a cone has radii $R = 10$ cm (larger base) and $r = 5$ cm (smaller base) and height $h = 12$ cm. Find its (a) volume and (b) total surface area.

Solution — Step by Step

Why This Works

A frustum is what’s left when a smaller cone is cut parallel to the base of a larger cone. The formulas come from subtracting the smaller cone’s volume/surface from the larger cone’s, then simplifying.

The slant height $l$ is the actual slant distance along the curved surface — NOT the vertical height $h$. Using $h$ instead of $l$ in the CSA formula is a common error that gives a wrong answer.

Alternative Verification — Check Slant Height

We can verify: if the big cone has height $H$ and radius $R$, and the cut is made at height $H - h$ from the apex, then:

$\frac{r}{R} = \frac{H-h}{H}$ gives the ratio. With $r = 5$, $R = 10$: the small cone is half the height of the big cone. So $H - h = h \Rightarrow H = 24$ cm.

Slant of big cone: $\sqrt{24^2 + 10^2} = \sqrt{676} = 26$ cm. Slant of small cone: $\sqrt{12^2 + 5^2} = 13$ cm. Slant of frustum = $26 - 13 = 13$ cm ✓

Common Mistake