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ryannakamura719
First find the area of the triangle (on top) and then subtract the area of the circle (on bottom) since the triangle is Equilateral, you can assume that the height can be found by using the 30 60 90 angle relationship. To find the radius of the circle, divide the height by 2.
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mhanifa

Answer:

(9√3 - 3π) cm²  or ≈ 6.16 cm²

Step-by-step explanation:

Refer to attached pictures

We have equilateral triangle with side of 6 cm and inscribed circle, we need to find the shaded area inside the triangle not covered by the circle.

The shaded are is the difference of areas of the triangle and circle

Area of equilateral triangle is calculated as per formula:

  • A= √3 /4 × a², where a is side of triangle
  • A= √3 /4 × 6² = 9√3 cm²

Area of circle is calculated as per formula:

  • A= πr², where r is the radius of circle

Let's find the value of r:

Δ ADB is the right triangle with angles 30° and 60°

It has sides of 3 cm, 6 cm and AD= m cm

As per attached, m is the long leg and equal to a√3, so

  • m = 3√3

also,

  • AD= AO+OD= AO +r

We can find AO in the same way as above using Δ AOF

AO= 2r as it is the hypotenuse and the hypotenuse is twice a short leg in the right triangle with angles 30° and 60°

So,  

  • AD= 2r+r= 3r
  • ⇒ r= AD/3 = 3√3/3= √3 cm

Now we can get the area of circle:

  • A= πr²= π×√3²= 3π cm²

Shaded are is:

  • (9√3 - 3π) cm²  or ≈ 6.16 cm²
Ver imagen mhanifa
Ver imagen mhanifa

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