# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/6 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #4 #. What is the area of the triangle's incircle?

The area of the triangle's incircle is

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The area of the incircle of a triangle can be calculated using the formula:

[ \text{Area of incircle} = \text{semiperimeter} \times \text{inradius} ]

The semiperimeter of the triangle can be calculated as ( s = \frac{a + b + c}{2} ), where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.

Given the angles at vertices A and B, and the area of the triangle, we can find the lengths of the sides using trigonometric relations.

Let's denote the side lengths as follows:

- ( a ) is opposite angle A,
- ( b ) is opposite angle B,
- ( c ) is opposite angle C.

We can use the law of sines to find the side lengths:

[ \frac{a}{\sin(\frac{\pi}{6})} = \frac{b}{\sin(\frac{\pi}{12})} = \frac{c}{\sin(\pi - \frac{\pi}{6} - \frac{\pi}{12})} ]

Then, once the side lengths are known, we can find the semiperimeter ( s ).

Next, we need to find the inradius ( r ). The formula to calculate the inradius of a triangle is:

[ r = \frac{\text{Area}}{\text{Semiperimeter}} ]

Finally, once we have the semiperimeter and inradius, we can find the area of the incircle using the formula mentioned at the beginning.

Let's proceed with the calculations:

Given: [ \text{Angle A} = \frac{\pi}{6} ] [ \text{Angle B} = \frac{\pi}{12} ] [ \text{Area of triangle} = 4 ]

We can use trigonometric relations to find the side lengths: [ a = 2 \times \sin(\frac{\pi}{6}) ] [ b = 2 \times \sin(\frac{\pi}{12}) ]

Then, calculate the semiperimeter: [ s = \frac{a + b + c}{2} ]

Find the inradius: [ r = \frac{\text{Area}}{s} ]

Finally, calculate the area of the incircle: [ \text{Area of incircle} = s \times r ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- A triangle has sides with lengths of 2, 7, and 6. What is the radius of the triangles inscribed circle?
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