SOLUTION: Find the radius of the circle inscribed in an equilateral triangle whose perimeter is 10.8 units

Algebra.Com
Question 1204108: Find the radius of the circle inscribed in an equilateral triangle whose perimeter is 10.8 units
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52800)   (Show Source): You can put this solution on YOUR website!
.
Find the radius of the circle inscribed in an equilateral triangle whose perimeter is 10.8 units.
~~~~~~~~~~~~~~~~~~~~~~~ 

For any triangle, from consideration its area, it is easy to deduce the formula 
for the radius "r" of the inscibed circle

     = area.    (1)

where P is its perimeter.


For the given equilateral triagle,  P = 3a,  area = ,  where "a" is the side length,

a =  = 3.6 units.


Therefore,  from (1)

     = .


It implies  r =  =  =  =  = 1.03923  (rounded).


ANSWER.  The radius of the inscribed circle is  r =  =  = 1.03923  (rounded).

Solved.

-------------------

For deducing formula  (1),  see the lesson
    - Proof of the formula for the area of a triangle via the radius of the inscribed circle
in this site.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

This is what the diagram could look like

A,B,C = vertices of the equilateral triangle
D = center of the inscribed circle
E = midpoint of AB


Triangle ADE is a right triangle. More specifically, it is a 30-60-90 triangle.
This is because angle CAB = 60 is bisected to help form angle DAE = 30 degrees.
Also, angle ADE = 60 degrees.

The perimeter of the equilateral triangle is 10.8 units.
Each side must be (10.8)/3 = 3.6 units
AB = 3.6
BC = 3.6
AC = 3.6

Because E is the midpoint of AB, we then know
AE = AB/2 = (3.6)/2 = 1.8

Segment AE is the longer leg of the 30-60-90 triangle ADE (notice it's opposite the 60 degree angle).

For any 30-60-90 triangle we have this template


In this particular case it means


Isolating DE gets us

Which is the approximate radius of the inscribed circle.

Round this value however needed. Or you can stick to the exact value to avoid worrying about rounding.

Side note: 0.6 = 3/5
RELATED QUESTIONS

An equilateral triangle with a perimeter of 24 square root (3) is inscribed in circle .... (answered by ewatrrr)
Find the perimeter of an equilateral triangle that can be inscribed in a circle whose... (answered by Fombitz)
If a circle is inscribed in an equilateral triangle of side 2v3 units, then radius of the (answered by Edwin McCravy)
what is the area of an equilateral triangle whose inscribed circle has radius... (answered by josgarithmetic)
an equilateral is inscribed in a circle of radius 10 cm. Find the equilateral's area. (answered by Alan3354)
An equilateral is inscribed in a circle of radius 10cm. Find the area of the equilateral. (answered by edjones)
An equilateral triangle is inscribed in a circle of radius 14cm. find the side of the... (answered by josgarithmetic)
The area of a circle inscribed in an equilateral triangle is 48s square units. Then the... (answered by solver91311)
An equilateral triangle is inscribed in a circle of radius 10cm , Find the area of... (answered by ikleyn)