Question

in a square of length 14 cm 4 quadrants are made inside the circle then find the area of the middle of all the segments

Answer 1

The green shaded area = Area of sector =60360πx2=60360πx2

=16πx2=16πx2

The blue shaded area = Area of segment = Area of sector - Area of equilateral triangle

=16πx2−3–√4x2=16πx2−34x2

A+E+D+H=blue+green=13πx2−3–√4x2A+E+D+H=blue+green=13πx2−34x2

A+E+D+H=13πx2−3–√4x2A+E+D+H=13πx2−34x2

A+E+D+H+C+G=A+E+D+H+C+G=Area of quarter circle  =14πx2=14πx2

A+E+D+H+C+G=14πx2A+E+D+H+C+G=14πx2

Subtraction the 2 equations,

C+G=3–√4x2−112πx2C+G=34x2−112πx2

Notice that, C+G=B+F=I+E=H+D=3–√4x2−112πx2C+G=B+F=I+E=H+D=34x2−112πx2

C+G+B+F+I+E+H+D=3–√x2−13πx2C+G+B+F+I+E+H+D=3x2−13πx2

Subtract this from area of square,

A=x2−(3–√x2−13πx2)A=x2−(3x2−13πx2)

A=x2−3–√x2+π3x2A=x2−3x2+π3x2

A=(1−3–√+π3)x2A=(1−3+π3)x2

A≈0.315147⋅x2A≈0.315147⋅x2

In your case, x=14x=14

A≈61.76876 cm2A≈61.76876 cm2