Question native Amar, a student:

I have actually been provided a circle inscribed in a triangle and have to be told come prove the the proportion of the perimeter the the triangle to the one of the one is the very same as the proportion of the area of the triangle to the area of the circle. Exactly how would this be done?

We have actually two responses because that you


Draw the circle inscribed in the triangle. Then attract radii to the clues of tangency. Then attract lines native the circle"s center to the 3 corners that the triangle.

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You have split the triangle right into 3 triangles and you can easily see the the height of each triangle is the radius. Due to the fact that the area that the triangle is the half the elevation times the base, friend can include the three smaller sized triangles" areas together to obtain the area of the big triangle.

Here"s a chart of what i mean:


So if the sides are A, B and also C and also the radius is R, then the Area of the triangle is RA/2 + RB/2 + RC/2.

Can you complete the difficulty from here?

Cheers, Stephen La Rocque.


It depends on what devices you have available for "thinking about" these connections.

(A) If you range the entire snapshot down through a factor K, then both areas scale by K^2, yet their ratios continue to be the constant. The scaling brings under the perimeters by a factor K, for this reason their proportion remains the same. If, by chance, you are in calculus, you might look at exactly how the areas are created by taking borders of slim strips the the perimeters, and also show that the proportion of the perimeters (A continuous in all the strips) becomes the ratio of the areas when you "add all the piece up".

I doubt you room not however in calculus - despite you might look because that this connection, later, when you are! This is true for various other shapes than triangles! This is in reality the huge idea what will lie behind every little thing other formulae girlfriend generate.

(B) Now exactly how to perform it "bare handed" together it were?

First - it help to break the triangle (and circle) down right into pieces. Here is one of the 6 pieces I would use.


If friend know about measures, you recognize m = r(theta) whereby theta is the edge In radians). You additionally know that k = r cotan(theta) but it turns out you yes, really don"t should know any of those formula!!

The comparison girlfriend really desire is the ratio of the space of the circle come the perimeter that the one is: (1/2) (mr)/ m = r/2. That is, because that this piece: AREA C = r/2(perimeter C)

The ratio of the area the the triangle to the perimeter contributed by the triangle is: (1/2) rk /k = r/2 AREA T = r/2 (perimeter T)

Now include up over all the 6 pieces. You will find the proportion of the area to to perimeter is tho r/2 end both the circle and also the triangle.

You have the right to take the from there.

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Walter Whiteley

Math central is sustained by the college of Regina and The Pacific Institute because that the mathematics Sciences.