SOLUTION: can you please help me solve and leave answer in terms of pie ((( two similar polygons have the scale factor 5:2. The area of the larger polygon is 100. Find the are of the smaller
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Question 198939This question is from textbook geometry
: can you please help me solve and leave answer in terms of pie ((( two similar polygons have the scale factor 5:2. The area of the larger polygon is 100. Find the are of the smaller polygon. )))
This question is from textbook geometry
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
A regular 
-gon has an area given by:
)
So if two similar regular polygons have a scale factor of 
then for a unit length of 
, the measure of the side of the smaller polygon is 
and the measure of the side of the larger is 
and the areas of the two -gons are:
^2\cot\left(\frac{\pi}{n}\right)=4\left(\frac{n}{4}k^2\cot\left(\frac{\pi}{n}\right)\right))
and
^2\cot\left(\frac{\pi}{n}\right)=25\left(\frac{n}{4}k^2\cot\left(\frac{\pi}{n}\right)\right))
So now we can say that:
But we are given that 
, so:
So:
In general, two similar -gons whose scale factor proportion is 
have areas that are in proportion 
. Using this general rule, the area proportion is 
which is equivalent to 
and 
.
Sorry, but 
has nothing to do with this. Shrug it off and go have a piece of cake.
John
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