# The radius of a circle is 10 cm. If the radius is increased by 20%, how do you find the percentage increase in area?

Feb 17, 2017

Solution given in a lot of detail so you can see where everything comes from.

Area increase is 44% of the original area

#### Explanation:

$\textcolor{b r o w n}{\text{Note that the % symbol is like a unit of measurement that is}}$$\textcolor{b r o w n}{\text{worth } \frac{1}{100}}$
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$\textcolor{b l u e}{\text{Setting up the initial condition and change}}$

20%" of "10 = 20/100xx10=2 larr" increase in radius"

Original area $\to \pi {r}^{2} = \pi {10}^{2} = 100 \pi$

New area $\to \pi {r}^{2} = \pi {12}^{2} = 144 \pi$

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$\textcolor{b l u e}{\text{Determine the percentage change}}$

Expressing the change as a fraction of the original area we have:

$\frac{144 \pi - 100 \pi}{100 \pi}$

Factor out the $\pi$ from $144 \pi - 100 \pi$ giving:

$\frac{\pi \left(144 - 100\right)}{\pi \times 100}$

This is the same as:

$\frac{\pi}{\pi} \times \frac{44}{100} \text{ "=" } 1 \times \frac{44}{100} = \frac{44}{100}$

This is the same as:

$44 \times \frac{1}{100}$

But $\frac{1}{100}$ is the same as % so we have:

44%