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

In the given figure,$ABCD$ is a square of side $14 \ cm$. $E$ and $F$ are mid-points of sides $AB$ and $DC$ respectively. $EPF$ is a semicircle whose diameter is $EF$. $LMNO$ is a square. What is the area (in $cm^2$) of the shaded region?

$A$ man walked diagonally across a square plot. Approximately what was the percent saved by not walking along the edges?

What would be the cost of laying a carpet on a floor which has its length and breadth in the respective ratio of $32:21$ and where its perimeter is $212$ feet,if the cost per square foot of laying the carpet is $₹ 2.5$?

The lengths of the two parallel sides of a trapezium are $16 \ m$ and $20 \ m$ respectively. If its height is $10 \ m$,its area in square metres is:

$A$ playground has the shape of a rectangle with two semi-circles on its smaller sides as diameters,added outside. If the sides of the rectangle are $36 \, m$ and $24.5 \, m$,then the area of the playground is ......... $m^2$ (use $\pi = \frac{22}{7}$).

Vedclass Test Series

Exam Paper Generator

Online Exam Module