#### To determine

**To find:** The total area occupied by the circles.

#### Answer

The total area occupied by the circles is,
11π96
.

#### Explanation

**Given:**

The triangle has sides of length 1.

**Proof:**

Consider the radius of the large circle is
r1
and radius of the next circle
r2
and so on.

From the above Figure 1,
∠BAC=60∘
.

cos60∘=r1|AB|12=r1|AB||AB|=2r1

From the Figure 1,
∠DBC=60∘

cos60∘=r2|DB|12=r2|DB||DB|=2r2

From the Figure 1,
|AB|=|AD|+|DB|

2r1=r1+r2+2r22r1=r1+3r2r1=3r2

In general,

3rn+1=rnrn+1=13rn

The total area *A* is sum of the area of circles.

A=πr12+3πr22+3πr32+⋯=πr12+3πr22+3π(13r2)2+3π(132r2)2⋯=πr12+3πr22(1+132+134+⋯)=πr12+3πr22⋅11−19 [∵a=1, r=19]

That is,

A=πr12+3πr22(98)

A=πr12+278πr22
(1)

Since the sides of the triangle have length 1, it can be concluded that
|BC|=12

From the Figure 1, in the angle
∠ABC
,
tan30∘=r1|BC|

tan30∘=r11213=2r1r1=123

If
r1=123
, then
r2=13(23)
.

That is,
r2=163

Substitute
r1 and r2
values in equation (1),

A=π(123)2+278π(163)2=π4⋅3+278⋅π36⋅3=π12+π32=8π+3π96

That is,
A=11π96
.

Hence, the total area occupied by the circles is
11π96
.