Concept 5

Solve for a missing dimension given area/surface area/volume information.

Use the area or volume formula to work backward from a given measure to the missing length.

Core Idea

When a problem gives you area or volume and asks for a length, set up the formula, plug in what you know, and solve for the missing dimension.

Understanding

Rule: Work backward from the given area or volume to the missing length.

  • Substitute the known values.
  • Isolate the variable.
  • Check whether the question wants the raw variable or a related value like diameter.

The algebra is usually simple; the key is reporting the right dimension.

Step by Step

  1. Write the relevant formula (area, surface area, or volume).
  2. Substitute every known value.
  3. Solve for the remaining variable.
  4. Re-read the question: does it ask for that variable, or something derived from it (like diameter from radius)?

Misconceptions

  • Finding the radius but reporting it as the diameter (or vice versa).
  • Forgetting to take the square root after isolating 𝑟2.
  • Solving for height when the question asks for slant height, which requires the Pythagorean theorem.
Question

Worked Example

A sphere has a volume of 2563𝜋 cubic inches. What is the diameter of the sphere, in inches?

Select an answer to see the explanation