Area and Volume

Use the perpendicular height in an area formula

A triangle with base 12 and perpendicular height 5 has area $A=\frac{1}{2}(12)(5)=30$ square units. A slanted side of length 13 is not the height unless it is perpendicular to the chosen base.

For a parallelogram, use $A=bh$ with the same perpendicular-height rule. For a rectangle, adjacent sides supply base and height because they meet at a right angle.

Match the solid to the volume formula

A cylinder of radius 3 and height 8 has volume $V=\pi r^{2}h=72\pi$ cubic units. If a problem instead gives diameter 6, divide by 2 before squaring.

A cone with the same radius and height has one-third that volume: $V=\frac{1}{3}\pi r^{2}h=24\pi$. The factor belongs to the solid's formula, not to its radius or height alone.

Use scale factors in the correct dimension

If every length of a figure is multiplied by 2, its area is multiplied by $2^2=4$ and its volume by $2^3=8$. If two similar solids have volumes in the ratio $27:8$, their length ratio is $3:2$.

The phrase "every length" matters. Doubling only a cylinder's height doubles its volume; doubling only its radius quadruples its volume. Do not apply a full similarity rule to a one-dimension change.

Keep exact values until the requested rounding

If the answer is requested in terms of $\pi$, leave $72\pi$ rather than replacing it with a decimal. If the question requests a rounded number, round after the final calculation.

Check units at the end. An area answer should use square units and a volume answer cubic units. A leftover length unit can reveal a missing factor in the setup.

Check your understanding

Sample area and volume questions

Practice this skill

1. Sketch and label the base, perpendicular height, and radius.
2. Write the formula before substituting numbers.
3. Include a scale-factor problem and a change-in-one-dimension problem.

Print the area and volume worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.