# Find the similarity ratio of two circles with areas 75 pi cm^2 and 27 pi cm^2.

Find the similarity ratio of two circles with areas $75\pi c{m}^{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}27\pi c{m}^{2}$.
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Step 1: Note down the given information
Area of the first circle, ${A}_{1}=75\pi c{m}^{2}$
Area of the second circle, ${A}_{2}=27\pi c{m}^{2}$
Let the radius of the circles be ${r}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{r}_{2}$ respectively.
Step 2: Calculate the ratio of areas of the the two circles
${A}_{1}:{A}_{2}=75\pi :27\pi$
$\pi$ cancels pi and we have both 27 and 75 divisible by 3
So, we can simplify this as
${A}_{1}:{A}_{2}=25:9$ ....(1)
Step 3: Calculate the ratio of areas in terms of ${r}_{1}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{r}_{2}$
The scale (similarity) factor for circles is the ratio of the radius
Similarity factor = ${r}_{1}:{r}_{2}$
Area of a circle = $\pi {r}_{2}$
so ${A}_{1}:{A}_{2}=\pi {r}_{1}^{2}:\pi {r}_{2}^{2}$
This simplifies to
${A}_{1}:{A}_{2}={r}_{1}^{2}:{r}_{2}^{2}$ .....(2)
Step 4: Use equation 1 and 2 to deduce similarity ratio
From 1 and 2, we have
${r}_{1}^{2}:{r}_{2}^{2}=25:9$
Taking square root both sides we get
${r}_{1}:{r}_{2}=5:3$
Result: So, the similarity ratio is 5 : 3