Question

On a circular table top of radius 30 cm a design is formed leaving an equilateral triangle inscribed in a circle. Find the area of the design. (???? = 3.14)

Answer 1

Do you think how to get area of the design ?
yes, we should find side length of equilateral triangle at first. and then area of design = area of circle - area of equilateral triangle .

see 2nd figure, Let ABC is and equilateral triangle inscribed in a circle of centre O.
then, AO = BO = CO = radius of circle = 30cm

as you know , angle between two sides of an equilateral be 60° .
so, ∠BAC = 60°
then, ∠BOC = 2∠BAC = 2 × 60° = 120°
now, from ∆BOC ,
∠OBC = ∠OCB ,[ because OB = OC = radius of circle ]
so, ∠OBC + ∠OCB + ∠BOC = 180°
=> 2∠OBC + 120° = 180°
=> ∠OBC = 30°

now, from ∆BOD,
cos∠OBC = BD/OB
cos30° = BD/r
BD = rcos30°
because , BD = DC = BC/2
so, BC = 2BD = 2rcos30°
where , r is the radius of circle,

now, BD = side length of ∆ABC = 2 × 30 × cos30°
= 2 × 30 × √3/2 = 30√3 m

now, area of ∆ABC= √3/4 × (side length)²
= √3/4 × (30√3)²
= √3/4 × 900 × 3
= 225 × 3√3 = 675√3 m² = 1167.75 m²

and area of circle = πr²
= π(30)² = 3.14 × 900
= 2826 m²

area of the design = 2826 - 1167.75
= 1658.25 m²