Area stacked between common tangent and circles

Question

Is there any way to find area of shaded region?

The radii of circles are $4$ and $12$ units.

Answer 1

You need three things: Thales, Pythagoras and trigonometry.

Let $O_1$ the center of the circle of radius $R_1$ and $O_2$ similarly.

Let's call $T$ the tangent.

Let's call $O$ the intersection between $O_1O_2$ and $T$.

Then Thales will give you $OO_1$ distance.

Pythagoras will give you then the lenght between $O$ and the circle $1$

Thales again for the height of the trapezoid...

Then you need to calculate the areas of the circles sections. Trigonometry is your friend.

Answer 2

Hint: BC // DE

Hint: Trapezoid - Sectors

Answer 3

By similar triangles, $CD=8$.

By Pythagoras' Theorem, $CG=EF=8\sqrt{3}$

Area of trapezoid $ACEF=\frac{(4+12)\times 8\sqrt{3}}{2}=64\sqrt{3}$

$\angle BAF=\cos^{-1} \left( \frac{1}{2} \right)=60^{\circ}$ and $\angle BCE=180^{\circ}-60^{\circ}=120^{\circ}$

$\therefore$ area of sector $BAF =\pi (12)^{2} \times \frac{60^{\circ}}{360^{\circ}} =24\pi$

$\therefore$ area of sector $BCE =\pi (4)^{2} \times \frac{120^{\circ}}{360^{\circ}} =\frac{16\pi}{3}$

The required area $=64\sqrt{3}-24\pi-\frac{16\pi}{3} =64\sqrt{3}-\frac{88\pi}{3}$