Three equal circles of unit radius touch one | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the centers of the three circles be $A, B,$ and $C$. Since each circle has a radius $r = 1$,the distance between any two centers is $AB = BC =

Vedclass · Accepted Answer

$\frac{\pi}{3}(2+\sqrt{3})^{2}$

Vedclass · Answer

(C) Let the centers of the three circles be $A, B,$ and $C$. Since each circle has a radius $r = 1$,the distance between any two centers is $AB = BC = CA = 2r = 2$. This forms an equilateral triangle $ABC$ with side length $s = 2$. The centroid $O$ of this triangle is the center of the circumscribing circle. The distance from the centroid to any vertex (the circumradius $R_{tri}$ of the triangle) is given by $R_{tri} = \frac{s}{\sqrt{3}} = \frac{2}{\sqrt{3}}$. The radius $R$ of the large circumscribing circle is the sum of the distance from the centroid to a vertex and the radius of one of the small circles: $R = R_{tri} + r = \frac{2}{\sqrt{3}} + 1 = \frac{2+\sqrt{3}}{\sqrt{3}}$. The area of the circumscribing circle is $\pi R^2 = \pi \left( \frac{2+\sqrt{3}}{\sqrt{3}} \right)^2 = \pi \frac{(2+\sqrt{3})^2}{3} = \frac{\pi}{3}(2+\sqrt{3})^2$.

Three equal circles of unit radius touch one another. Then the area of the circle circumscribing the three circles is

Similar Questions

$A$ cuboid of dimension $24 \text{ cm} \times 9 \text{ cm} \times 8 \text{ cm}$ is melted and smaller cubes of side $3 \text{ cm}$ are formed. Find how many such cubes can be formed?

The area of a circle is $154 \, cm^{2}$. The length of an arc of the circle which subtends an angle of $45^{\circ}$ at the centre is......$cm$.

The perimeter of a circle is equal to that of a square. Compare their areas.

The areas of a square and a rectangle are equal. The length of the rectangle is greater than the side of the square by $5 \text{ cm}$ and the breadth is less than the side of the square by $3 \text{ cm}$. Find the perimeter of the rectangle (in $\text{cm}$).

Vedclass Test Series

Exam Paper Generator

Online Exam Module