In a reverse-dimension problem you are told the volume and must find a missing length. Rearrange the formula first, then substitute: for a cylinder, h = V ÷ πr², and r = √(V ÷ πh).
This lesson uses everything before it in volume and capacity, especially cylinders and cones and unit conversion.
Why rearrange before substituting?
Substituting first leaves one number with a mess of multiplications around it. Rearranging first gives you a clean, reusable line that shows exactly what to divide by.
Think of the formula as a chain of multiplications. To isolate one link, undo the others by dividing.
How to work through a question
- Write the volume formula for the solid.
- Rearrange to make the missing length the subject.
- Substitute the known values, converting units if needed.
- Calculate, using the full calculator value.
- If you found r², take the square root.
- Round at the end and check the answer is sensible for the shape.
Worked example
A cylinder has height 8 cm and volume 450 cm³. Find its radius, to 3 significant figures.
Step 1, formula: V = πr²h.
Step 2, rearrange: r² = V ÷ (πh).
Step 3, substitute: r² = 450 ÷ (π × 8) = 450 ÷ 25.13… = 17.90…
Step 4, square root: r = √17.90… = 4.231…
So the radius is 4.23 cm.
Check by going forward: π × 4.23² × 8 = π × 17.89 × 8 = 449.7, which is close to 450, as expected from rounding.
The mistake to watch for
The typical slip is to stop at r².
Mistaken working: r² = 17.9, so r = 17.9 cm
The value 17.9 is the radius squared, not the radius.
The correction is to take the square root as the final step, giving 4.23 cm. A sense check also catches this: a radius of 17.9 cm would need a far bigger volume than 450 cm³ for a height of only 8 cm.
Check yourself
1. A cuboid has volume 360 cm³, length 10 cm and width 6 cm. Find its height.
Show answer
Height = 360 ÷ (10 × 6) = 360 ÷ 60 = 6 cm.
2. A cone has volume 100 cm³ and base radius 4 cm. Find its vertical height to 3 significant figures.
Show answer
V = ⅓πr²h, so h = 3V ÷ (πr²) = 300 ÷ (π × 16) = 300 ÷ 50.26… = 5.968… so 5.97 cm.
3. A cylinder has volume 1000 cm³ and radius 5 cm. Find its height to 3 significant figures.
Show answer
h = 1000 ÷ (π × 25) = 1000 ÷ 78.53… = 12.73… so 12.7 cm.
Where this leads next
The last lesson in the module is estimating whether a container result is plausible, a habit that would catch the missing square root above. The non-calculator working trainer helps with the exact arithmetic when a question allows it.
Rearranging joins geometry to algebra, and many students find both easier once a teacher links them. Online one-to-one Mathematics tuition can focus on exactly that step.