Right Circular Cylinder

A right circular cylinder is the 3D shape you get by rotating a rectangle about one of its sides , or equivalently, the shape of a soup can. Two parallel circular bases joined by a curved rectangular "label" wrapped around them. This lesson gives all the relevant formulas.

Definition

A right circular cylinder has two congruent parallel circular bases of radius $r$, joined by a curved surface. The perpendicular distance between the bases is the height $h$. The line from the centre of one base to the centre of the other is the axis , perpendicular to both bases (hence "right" cylinder).

Formulas

For a cylinder of radius $r$ and height $h$:

• Curved surface area (CSA): $2\pi r h$.
• Total surface area (TSA): $2\pi r h + 2 \pi r^2 = 2\pi r(h + r)$.
• Volume: $V = \pi r^2 h$.

Why the formulas are right

Curved surface. Imagine cutting the curved surface vertically and unrolling it. You get a rectangle of width $2\pi r$ (the circumference of the base) and height $h$. Area: $2\pi r h$.

Total surface. Add the two circular bases, each of area $\pi r^2$. Total: $2\pi r h + 2 \pi r^2 = 2\pi r(h + r)$.

Volume. The base area is $\pi r^2$; stacked through height $h$. Volume: $\pi r^2 h$. (Just like cuboid, but with a circular base.)

Hollow cylinders

A hollow cylinder (pipe) has an outer radius $R$ and inner radius $r$, with the same height $h$.

• Volume of material: $\pi (R^2 - r^2) h$.
• Total surface area is more complex , includes the outer curved surface, inner curved surface, and the two ring-shaped ends.

Worked examples

Example 1. A cylinder has radius $7$ cm and height $10$ cm. Find CSA, TSA, V. (Use $\pi = \tfrac{22}{7}$.)

$\text{CSA} = 2 \cdot \tfrac{22}{7} \cdot 7 \cdot 10 = 440$ cm$^2$. $\text{TSA} = 2\pi r(h + r) = 2 \cdot \tfrac{22}{7} \cdot 7 \cdot 17 = 748$ cm$^2$. $V = \pi r^2 h = \tfrac{22}{7} \cdot 49 \cdot 10 = 1540$ cm$^3$.

Example 2. A cylindrical water tank has diameter $14$ m and height $20$ m. Find its capacity.

Radius $= 7$ m. $V = \tfrac{22}{7} \cdot 49 \cdot 20 = 3080$ m$^3$ $= 3080000$ litres.

Example 3. The CSA of a cylinder is $176$ cm$^2$ and base radius is $7$ cm. Find the height.

$\text{CSA} = 2\pi r h = 2 \cdot \tfrac{22}{7} \cdot 7 \cdot h = 44 h = 176 \Rightarrow h = 4$ cm.

Example 4. A cylindrical pillar of diameter $50$ cm and height $3$ m is to be painted. Find the cost at Rs. $20$ per m$^2$.

Radius $= 25$ cm $= 0.25$ m. $\text{CSA} = 2\pi r h = 2 \cdot \tfrac{22}{7} \cdot 0.25 \cdot 3 = \tfrac{33}{7}$ m$^2$ $\approx 4.71$ m$^2$. Cost $\approx 4.71 \cdot 20 \approx$ Rs. $94.29$.

Example 5. A hollow cylindrical pipe has outer radius $5$ cm, inner radius $4$ cm, and length $20$ cm. Find the volume of material.

Volume $= \pi(R^2 - r^2) h = \pi(25 - 16) \cdot 20 = 180\pi \approx 565.71$ cm$^3$.

Try it yourself

1. Cylinder: $r = 5, h = 12$. Find CSA, TSA, V.
2. Cylinder: diameter $14$, height $15$. Find volume (use $\pi = \tfrac{22}{7}$).
3. A cylindrical container has radius $7$ cm and height $20$ cm. How many litres can it hold? ($1$ litre $= 1000$ cm$^3$.)
4. CSA of a cylinder of radius $10$ is $440$ cm$^2$. Find the height.
5. A pipe is hollow, outer radius $7$, inner radius $5$, length $14$. Find the volume of material.
6. Cylinder of radius $r$ and height $r$. State CSA, TSA, V.
7. A cylindrical pillar is $4$ m high with circumference $44$ cm. Find the cost of painting at Rs. $5$/m$^2$ for the curved part.
8. A cylindrical tank of radius $1.4$ m holds $19250$ litres. Find the height. ($1$ m$^3$ $= 1000$ litres.)
9. A cylinder has total surface area $968$ cm$^2$ and height $10$. Find the radius.
10. Express the volume of a cylinder in terms of its CSA and radius.

Pitfalls / Insight

• Diameter vs. radius. Always halve the diameter to get the radius.
• CSA vs. TSA. CSA covers only the wrap-around; TSA adds the two circular bases.
• Units. Convert all measurements to a single unit before computing.

Insight. Think of a cylinder as "a rectangle rolled into a cylinder, plus two circles". This visualisation alone gives you all the surface-area formulas.