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Frustum of a Cone

Surface Areas and Volumes · Class X

Slide the two radii and height of a frustum; read slant height, CSA, TSA, and volume.

6
type a value
0.515
3
type a value
0.115
8
type a value
0.520
Live values
  • Slant height l8.544
  • Curved surface area241.576
  • Total surface area382.9477
  • Volume527.7876
xy
  • Volume vs height (R, r fixed)

Formulas in this lab

  • Slant height l
    h2+(R−r)2\sqrt{h^2 + (R-r)^2}
  • Curved surface area
    π(R+r)l\pi(R+r)l
  • Total surface area
    π[(R+r)l+R2+r2]\pi[(R+r)l + R^2 + r^2]
  • Volume
    13πh(R2+r2+Rr)\frac{1}{3}\pi h(R^2 + r^2 + Rr)
Tip: Volume grows linearly with height; the slope depends on R^2 + r^2 + R*r.

Frequently asked questions

▶What is a frustum and its formulas?

A frustum is a cone with the top sliced off parallel to the base. With radii R and r and height h, slant $l = \sqrt{h^2 + (R-r)^2}$, $CSA = \pi(R + r)l$, $TSA = \pi(R + r)l + \pi R^2 + \pi r^2$, and $V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)$.

▶How do I use this lab?

Slide R, r and h. The lab shows l, CSA, TSA and V together. Try R = 6, r = 3, h = 8 to see how the slant works out and how volume sits between a full cone and a cylinder.

▶Common mistake on this topic

Students drop the $Rr$ term in the volume formula, writing $R^2 + r^2$ instead of $R^2 + r^2 + Rr$. They also use the cone slant $\sqrt{r^2 + h^2}$ instead of $\sqrt{h^2 + (R-r)^2}$ for the frustum.

▶Where do we see frustums?

A bucket, a kulhad (clay tea cup), and a lampshade are all frustums. Knowing the volume tells you how many litres your kitchen bucket holds; the lab lets you check before buying a new one.