A regular octagon is circumscribed by a circle of radius rem. find the area enclosed between the circle and the octagon (give the answer in terms of r.)
1. The answer is 0.313r².
2. Show me the maths step by step.
3. Whoever, shows the math step by step gets to marked as "brainlist".
Answers
In the question, we are given a regular octagon with a circle of radius r circumscribing the octagon. We first draw the figure and then analyze the problem. So, we have to find the area of the regular octagon inscribed in a circle of radius r.
Hence, length of AO = radius of circle = r
Similarly, length of BO = radius of circle = r
Regular octagon has all sides equal.
Thus angle subtended by each edge of regular octagon at centre of the circle is equal
So, ∠AOB=360∘8=45∘
Now, area of ΔAOB = 12r2sinθ
=12r2sin45∘
=12r212–√
=r222–√
Now, to find the area of a regular octagon, we have to find areas of all the eight triangles separately and add them all.
Since all the triangles thus formed are congruent to each other with measure of all the sides being equal. So, the area of all the triangles would be equal.
Area of Octagon = Area of 8 triangles = 8× ( area of ΔAOB )
Thus, area of octagon =8r222–√
=4r22–√
=22–√r2
Thus, the area of regular octagon inscribed in a circle of given radius r =22–√r2
So, the correct answer is “ =22–√r2 ”.
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