A triangle has vertices A, B, and C. Vertex A has an angle of $pi/8$ , vertex B has an angle of $( pi)/6$ , and the triangle's area is $3$ . What is the area of the triangle's incircle?

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A triangle has vertices A, B, and C. Vertex A has an angle of $pi/8$ , vertex B has an angle of $( pi)/6$ , and the triangle's area is $3$ . What is the area of the triangle's incircle?

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Answer 1 · 2017-06-26T09:30:40

The area of the incircle is $=1.02$ square units.

Explanation:

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The area of the triangle is $A=3$

The angle $hatA=1/8pi$

The angle $hatB=1/6pi$

The angle $hatC=pi-(1/8pi+1/6pi)=17/24pi$

The sine rule is

$a/sinA=b/sinB=c/sinC=k$

So,

$a=ksinA$

$b=ksinB$

$c=ksinC$

Let the height of the triangle be $=h$ from the vertex $A$ to the opposite side $BC$

The area of the triangle is

$A=1/2a*h$

But,

$h=csinB$

So,

$A=1/2ksinA*csinB=1/2ksinA*ksinC*sinB$

$A=1/2k^2*sinA*sinB*sinC$

$k^2=(2A)/(sinA*sinB*sinC)$

$k=sqrt((2A)/(sinA*sinB*sinC))$

$=sqrt(6/(sin(pi/8)*sin(1/6pi)*sin(17/24pi)))$

$=6.29$

Therefore,

$a=6.29sin(1/8pi)=2.41$

$b=6.29sin(1/6pi)=3.15$

$c=6.29sin(17/24pi)=4.99$

Observe in the figure above that joining incenter with the three vertices $A,B$ and $C$ results in three triangles, whose bases are $a,b$ and $c$ and height is $r$ , the radius of incenter and therefore the sum of the area of these triangles is area of $DeltaABC$ .

Hence we have, $1/2*r*(a+b+c)=A$

$r=(2A)/(a+b+c)$

$=6/(10.55)=0.57$

The area of the incircle is

$area=pi*r^2=pi*0.57^2=1.02$ square units.

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A triangle has vertices A, B, and C. Vertex A has an angle of $pi/8$ , vertex B has an angle of $( pi)/6$ , and the triangle's area is $3$ . What is the area of the triangle's incircle?

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

A triangle has vertices A, B, and C. Vertex A has an angle of pi/8 pi/8 , vertex B has an angle of ( pi)/6 ( pi)/6 , and the triangle's area is 3 3 . What is the area of the triangle's incircle?

1 Answer

Explanation:

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Impact of this question

A triangle has vertices A, B, and C. Vertex A has an angle of $pi/8$ , vertex B has an angle of $( pi)/6$ , and the triangle's area is $3$ . What is the area of the triangle's incircle?