Find the volume of a cylinder with radius 3 and height 10.

Step-by-step solution

1. 1

   Understand the formula as area times height. A cylinder is a circle swept straight up, so its volume is the area of the circular base multiplied by the height: $V=\pi r^2h$. Every prism-like solid follows this same base-times-height pattern.

2. 2

   Compute the base area. With $r=3$, the base area is $\pi r^2=\pi(3)^2=9\pi$ square units. Squaring the radius first, before multiplying by anything else, avoids the common slip of computing $(\pi r)^2$.

3. 3

   Multiply by the height. $V=9\pi\times 10=90\pi$ cubic units. This exact form is preferred whenever the question does not force a decimal.

4. 4

   Give the decimal value. Using $\pi=3.14159265$, $V=90\times 3.14159265=282.74$ cubic units to two decimal places. (Using the coarser $\pi\approx 3.14$ gives $282.6$, which is why the exact form is safer.)

5. 5

   Check the order of magnitude. The cylinder fits inside a box $6\times 6\times 10=360$ cubic units, and a circle fills $\frac{\pi}{4}\approx 0.7854$ of its bounding square: $0.7854\times 360=282.74$, matching the computed volume exactly.