Question

Find the radius of the circle circumscribed around an equilateral triangle, if the radius of the circle inscribed into this triangle is 10 cm.

Answer 1

20 cm

Explanation:

Look at the picture.

The formula of a radius of a circle circumscribed around an equaliteral triangle:

`R=(2)/(3)\cdot(a\sqrt3)/(2)`

The formula od a radius of a circle inscribed into an equaliteral triangle:

`r=(1)/(3)\cdot(a\sqrt3)/(2)`

As you can see in the formulas above, the radius of the circumscribed circle is twice the radius of the inscribed circle.

Therefore

`R=2r`

Given:

`r=10\ cm`

therefore

`R=2(10\ cm)=20\ cm`

Answer 2

20 cm

Explanation:

Let a cm be the length of the side of equilateral triangle.

Use formula for the radius of inscribed circle into the equailteral triangle:

`r_(inscribed)=(a√(3))/(6)`

Hence,

`(a√(3))/(6)=10\Rightarrow a=(60)/(√(3))`

Now, use formula for the circumscribed circle's radius:

`R_(circumscribed)=(a√(3))/(3)`

Therefore,

`R_(circumscribed)=((60)/(√(3))\cdot √(3))/(3)=20\ cm`