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: An approximation is a value deliberately chosen near the true one, while tracking how far off it could be.

Common stuck point: The procedure for approximation is the easy part; the trap is forgetting the error exists. Asking "Am I deliberately using a near value for a hard-to-write exact one while caring how far off it is?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I deliberately using a near value for a hard-to-write exact one while caring how far off it is?

Worked Examples

Example 1

easy

Estimate

\sqrt{50}

to one decimal place without a calculator.

Answer

\sqrt{50} \approx 7.1

First step

Identify perfect squares on either side:

7^2 = 49

and

8^2 = 64

. So

7 < \sqrt{50} < 8

Full solution

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
To one decimal place: $\sqrt{50} \approx 7.1$ .

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

Example 3

medium

Estimate

\sqrt{40}

to one decimal place without a calculator.

Example 4

medium

Estimate

999^2

using the identity

(a - 1)^2 = a^2 - 2a + 1

with

a = 1000

. Then compute exactly.

Example 5

medium

Use

\sqrt{10} \approx 3.162

to estimate

\sqrt{1000}

and

\sqrt{0.1}

Example 6

hard

Use the linear approximation

(1 + x)^n \approx 1 + nx

for small

x

to estimate

1.02^{10}

Example 7

hard

Estimate

\sqrt{82}

using the linear approximation

\sqrt{a + \epsilon} \approx \sqrt{a} + \dfrac{\epsilon}{2\sqrt{a}}

with

a = 81

Example 8

hard

Estimate

\sin(0.1)

using

\sin x \approx x - x^3/6

for small

x

in radians.

Example 9

challenge

A surveyor estimates a hill's height using the angle of elevation. If the measured angle is

30.0^\circ \pm 0.5^\circ

and the horizontal distance is

200

m (exact), give the approximate height and its uncertainty.

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}

Example 3

easy

Round

\pi

to two decimal places.

Example 4

easy

Estimate

48 + 53

by rounding each to the nearest ten.

Example 5

easy

\sqrt{2} = 1.414

exact?

Example 6

easy

Round

6849

to the nearest hundred.

Example 7

easy

Approximate

\frac{22}{7}

as a decimal to two places.

Example 8

easy

Estimate

\sqrt{50}

to the nearest whole number.

Example 9

easy

True or false: every approximation has some error.

Example 10

easy

Round

0.0467

to two decimal places.

Example 11

medium

Using

\pi \approx 3.14

, find the area of a circle with radius

5

. Note the result is approximate.

Example 12

medium

Estimate

19.8 \times 5.1

using rounding, then state how it compares to the exact value.

Example 13

medium

The error of using

3.14

for

\pi

is about

0.0016

. Estimate the area error for a circle of radius

10

Example 14

medium

Approximate

\frac{1}{3}

to three decimal places and state the rounding error.

Example 15

medium

Round

4.5

and

5.5

to the nearest whole number using round-half-up.

Example 16

medium

A budget of

\$2980

is rounded to

\$3000

for a report. What is the rounding error, and is it an over- or under-estimate?

Example 17

medium

To estimate

\sqrt{20}

between consecutive tenths, which two values bracket it?

Example 18

challenge

Using

\pi \approx 3.14

\pi \approx 3.14159

, compute the circumference of a circle with radius

100

both ways and find the difference.

Example 19

challenge

A number

x

rounds to

3.5

at one decimal place. Give the exact interval of possible values of

x

Example 20

challenge

Estimate

\sqrt{2} \approx 1.4

and use one Newton step

x_1 = \frac{1}{2}(x_0 + \frac{2}{x_0})

to improve it.

Example 21

medium

Estimate

\frac{612}{29}

to the nearest whole number.

Example 22

medium

A measurement is

25

m with an estimated error of

\pm 1

m. Express this as a percent error.

Example 23

easy

Approximate

\sqrt{17}

to the nearest whole number.

Example 24

easy

Estimate

\sqrt{83}

to the nearest whole number.

Example 25

easy

True or false:

\sqrt{2} \approx 1.414

has zero error.

Example 26

easy

Estimate

19 \times 21

by rounding to nice numbers, and compare to the exact value.

Example 27

medium

Approximate the cost of

7

items at $3.97 each.

Example 28

medium

Compute the percent error when

\pi

is approximated as

3.14

(use the true value

3.14159

Example 29

medium

Approximate

1{,}492 \div 7

to the nearest hundred.

Example 30

medium

Estimate

48 \times 19

using rounding.

Example 31

hard

Use

\pi \approx 22/7

to estimate the circumference of a circle of radius

21

cm. Find the percent error vs

\pi \approx 3.14159

Example 32

hard

Approximate

\ln(1.05)

using

\ln(1+x) \approx x - x^2/2

for small

x

Example 33

hard

A circular table has measured diameter

1.20

m (with

\pm 0.005

m uncertainty). Approximate its area and its uncertainty.

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Related Concepts

Estimation Irrational Numbers Error Analysis Limit

Background Knowledge

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

estimationirrational numbers