Question

What is the ratio of areas of circum circle and in circle of an equilateral triangle?

Answer 1

Given
Δ
A
B
C
=
equilateral triangle
Let radius of in-circle be
r
, and radius of circumcircle be
R
.
In
Δ
O
B
D
,
∠
O
B
D
=
30
∘
,
∠
O
D
B
=
90
∘
⇒
R
=
2
r
Let area of in-circle be
A
I
and area of circumcircle be
A
C
,
⇒
A
I
A
C
=
π
r
2
π
(
R
2
)
=
π
r
2
π
(
2
r
)
2
=
1
4
Hence,
A
I
:
A
C
=
1
:
4