3 circle of radius 1 cm are drawn taking vertex a centre in strangle of ide 3, 4, 6, cm each find the area of a circle remained outside triangle
Answers
Step-by-step explanation:
Given, A,B and C are the centres of three circles touching each other at P,Q& R. All the circles have equal radii r=3.5 cm.
To find out: The area enclosed by the circles.
Solution:
We join AB,BC and CA. Since AB joins the centres of two circles touching each other, we have AB= sum of their radii =2r=2×3.5 cm =7 cm.
Similarly BC=7 cm and CA=7 cm.
∴ΔABC is equilateral with sides =a=7 cm.
So area ΔABC=
4
3
×a
2
=
4
3
×7
2
cm
2
=
4
49
3
×cm
2
=21.217 cm
2
.
Also each angle of ΔABC=60
o
.
Now AQ&AP are the radii of the same circle of which
PQ is an arc.
∴APQ is a sector of central angle =θ=60
o
and radius =r=3.5 cm.
∴ Area of sector APQ=
360
o
θ
×π×r
2
=
360
o
60
o
×
7
22
×3.5
2
cm
2
=6.417 cm
2
.
Since all the sectors have same radii and central angle they are equal in areas.
∴ Area of three sectors =3×6.417 cm
2
=19.251 cm
2
.
∴ Area of enclosed region = area .ΔABC− Area of three sectors =(21.217−19.251)cm
2
=1.966 cm
2