Area of Circle | Circle inside Quarter Circle and Square problem | Advanced math problems | Geometry

Usually tangents are referred to tangent lines to a circle or a curve at some point. But in fact if two circles are touching each other they are called tangent circles, and they have many interesting properties. This problem is an application of tangent circles.

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#maths #mathematics #circle #square #quartercircle #area #tangent #advancedmathproblems #calculation #problems #geometry

[miguelgnievesl]

If the center of the circle is O and r is the radius, then OP=OR=RC=r. The right triangle ORC is formed and applying the Pythagorean Theorem we obtain OC= r√2. The radius of the circular sector is PQ=PO+OC=r+r√2= r(√2+1)=5. Consequently r=5/(√2+1) or r=5(√2-1) or r=5√2-5. Finally, the area of the circle is A(cir)=πr²=π(5√2-5)²= π(75-50√2) o A(cir)=25π(3-2√2).

[ludosmets]

In the little square with sides r the diagonal = 5-r. So, 2r^2= (5-r)^2 which gives the same quadratic formula r^2+ 10r-25=0

[Grizzly]

Too long winded, and no need to go anywhere near a quadratic equation.
Just using tangent circle theorems and Pythagoras, you should be able to easily arrive at r + r√2 = 5 and it should be plain sailing from there.