Centred at each corner of a regular hexagon, a part of a circle is drawn and a figure is cut out as shown below:

What is the area of this figure?

As seen from the figure,

Radius of each arc =

For a regular polygon with n sides,

Sum of angles = (n-2)× 180°

Here, n = 6

∴ Sum of angles = (6-2)× 180

= 4× 180

= 720°

Hence, each angle =

Thus for each sector,

X = 120°

r = 1 cm

If x° is the angle at subtended at the sector,

Area of sector = r²

∴ Area of sector = 1² = cm²

Area of 6 sectors = cm²

As the hexagon is made up of 6 equilateral triangles,

Area of polygon = 6× Area of equilateral triangle with side as 2 cm

If a = side

Then area of triangle = (side)²

Here, side = 2 cm

∴ area of a triangle = (2)² = cm²

∴ area of polygon = 6× √3 = 6√3 cm²

∴ area of shaded region = area of polygon-area of 6 sectors

= 6√3 – 2π = 4.112 cm²