ABC is an equilateral triangle inscribed in a circle of radius 4cm with
centre O. Find the area of the shaded region.
Answers
Answer:
Answer:
Area=\frac{16\pi}{3}-4\sqrt{3}Area=
3
16π
−4
3
Step-by-step explanation:
In the given figure, we have a circle and an equilateral triangle,
Radius of the circle, r = 4 cm
Triangle is equilateral.
So,
Length of the side of the triangle, is given by,
AC = 2OA.cos30°
AC = 2r.cos30°
AC = 8.cos30°
AC = 4√3 cm
Now,
Area of the circle is given by,
Area=\pi r^{2}=\pi(4)^{2}=16\pi\, cm^{2}Area=πr
2
=π(4)
2
=16πcm
2
Now,
Area of the Equilateral triangle is given by,
\begin{gathered}Area=\frac{\sqrt{3}}{4}a^{2}\\Area=\frac{\sqrt{3}}{4}(4\sqrt{3})^{2}\\Area=12\sqrt{3}\, cm^{2}\end{gathered}
Area=
4
3
a
2
Area=
4
3
(4
3
)
2
Area=12
3
cm
2
Therefore, the Area of the Shaded Region is given by,
Area = Area of circle - Area of Triangle
So,
\begin{gathered}Area=\frac{16\pi-12\sqrt{3}}{3}\\Area=\frac{16\pi}{3}-4\sqrt{3}\end{gathered}
Area=
3
16π−12
3
Area=
3
16π
−4
3
Therefore, the Area of the shaded region is given by,
Area=\frac{16\pi}{3}-4\sqrt{3}Area=
3
16π
−4
3