Approximation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Approximation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A value intentionally chosen to be close to but not exactly equal to the true value, with a known or estimated error.

We use 3.14 for \pi, knowing it's not exactly right but close enough.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: All measurements and many calculations give approximations, not exact values.

Common stuck point: Knowing how good an approximation is—always check the error: |\text{approx} - \text{true}| gives the absolute error.

Sense of Study hint: Compute the difference between your approximation and the exact value (or a better approximation) to see how much error you introduced.

Worked Examples

Example 1

easy
Estimate \sqrt{50} to one decimal place without a calculator.

Solution

  1. 1
    Identify perfect squares on either side: 7^2 = 49 and 8^2 = 64. So 7 < \sqrt{50} < 8.
  2. 2
    \sqrt{50} is very close to \sqrt{49} = 7. The gap between 49 and 64 is 15; 50 is 1 above 49, so \sqrt{50} \approx 7 + \dfrac{1}{15} \approx 7.07.
  3. 3
    To one decimal place: \sqrt{50} \approx 7.1.

Answer

\sqrt{50} \approx 7.1
Linear interpolation between known square roots gives a reasonable first approximation. Knowing the perfect squares on either side of the target immediately brackets the root, and the fraction of the gap gives the tenths digit.

Example 2

medium
Use the approximation \pi \approx \dfrac{22}{7} to estimate the circumference and area of a circle with radius 14 cm. Compare with the decimal approximation \pi \approx 3.14159.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Estimate 198 \times 52 using rounding, then compute the exact answer and find the percent error.

Example 2

medium
Use the first-order linear approximation \sqrt{a+\epsilon} \approx \sqrt{a} + \dfrac{\epsilon}{2\sqrt{a}} to estimate \sqrt{101}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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