what is the largest sqare that can be incribed in a circle of radius R
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The area of the largest square that can be inscribed in a semicircle is (4r²)/5 , where r is the radius of the semicircle.
Let's suppose that b is the largest possible side of the square that can be inscribed in a semicircle. So one side PQ of the square will be lying on diameter DB. And centre O will be the mid point of PQ.
As shown in the above fig no 1 . PQRS square is inscribed in semicircle DBRS. Its side SP = PQ = b unit. And half of PQ (b/2) lies equally on either side of the centre of the circle O.
In right triangle SPO , b² + (b/2)² = r² ( by Pythagoras law)
=> b² + b²/4=r²
=> 5b²/4 = r²
=> b² = 4r²/5
So, AREA of the Square = b²= 4r²/5
Let's suppose that b is the largest possible side of the square that can be inscribed in a semicircle. So one side PQ of the square will be lying on diameter DB. And centre O will be the mid point of PQ.
As shown in the above fig no 1 . PQRS square is inscribed in semicircle DBRS. Its side SP = PQ = b unit. And half of PQ (b/2) lies equally on either side of the centre of the circle O.
In right triangle SPO , b² + (b/2)² = r² ( by Pythagoras law)
=> b² + b²/4=r²
=> 5b²/4 = r²
=> b² = 4r²/5
So, AREA of the Square = b²= 4r²/5
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