Question

The base area of a cylinder is \(154\ \text{cm}^2\) and the height is \(5\ \text{cm}\). Find the volume of the cylinder.

Hint:

If the base area is given, use \(V=\text{base area}\times \text{height}\) directly. If only the radius is given, first compute the base area using \(\pi r^2\), then multiply by the height. Always confirm units: \(\text{area}\times\text{length} \text{volume}\).

Answer 1

Step 1 (Recall the definition of volume for a cylinder).
For any right circular cylinder, the volume is the product of the base area and the vertical height:
\[ V \;=\; \text{(area of circular base)} \times \text{(height)}. \] This is the same as \(V=\pi r^2 h\), but if the base area is already given, we can use it directly. Method 1 — Using the given base area directly.
Step 2 (Substitute values with units).
Given base area \(A_b = 154\ \text{cm}^2\) and height \(h=5\ \text{cm}\):
\[ V \;=\; A_b \times h \;=\; 154\ \text{cm}^2 \times 5\ \text{cm}. \] Step 3 (Multiply and track units).
\[ V \;=\; 770\ (\text{cm}^2\cdot \text{cm}) \;=\; 770\ \text{cm}^3. \] The unit \(\text{cm}^2 \times \text{cm} \text{cm}^3\) confirms we have a volume. Method 2 — Cross-check by recovering the radius.
Step 4 (Relate base area to radius).
For a circle, \(A_b=\pi r^2\). With \(A_b=154\ \text{cm}^2\) and taking \(\pi=\dfrac{22}{7}\) (a standard school value):
\[ \pi r^2=154 r^2=\frac{154}{\pi}=\frac{154}{22/7}=154\cdot\frac{7}{22}. \] Step 5 (Simplify exactly).
Note that \(154=22\times 7\). Hence
\[ r^2=\frac{22\cdot 7\cdot 7}{22}=49 r=7\ \text{cm}. \] Step 6 (Compute volume via } \(\pi r^2 h\){).
\[ V=\pi r^2 h=\frac{22}{7}\times 7^2 \times 5=\frac{22}{7}\times 49 \times 5 =\frac{22}{7}\times 245=22\times 35=770\ \text{cm}^3. \] This exactly matches Method 1. Step 7 (Sanity/size check).
A base of \(154\ \text{cm}^2\) is about a circle of radius \(7\ \text{cm}\). A height of \(5\ \text{cm}\) is modest, so a volume slightly under \(1000\ \text{cm}^3\) (one liter) is reasonable. Our \(770\ \text{cm}^3\) fits this intuition. \[ {770\ \text{cm}^3} \]