An equilateral triangle is inscribed in a circle with a radius of 6". Find the area of the segment cut off by one side of the triangle.
To find the area of the segment cut off by one side of the equilateral triangle, we can follow these steps:
1. Find the area of the equilateral triangle:
Since the triangle is equilateral, all three sides are equal. Let's call the side length "s".
The formula for the area of an equilateral triangle is A = (sqrt(3) / 4) * s^2.
In this case, we know that the radius of the circle is 6 inches, so the diameter is 12 inches.
The diameter of the circle is also equal to the length of the side of the equilateral triangle.
Hence, s = 12 inches.
Now, substitute the value of "s" into the formula for the area of an equilateral triangle to find the area of the triangle.
2. Find the area of the sector formed by the triangle:
The triangle divides the circle into six equal sectors (since it is equilateral).
To find the area of one sector, we can use the formula for the area of a sector: A = (θ / 360) * π * r^2.
In this case, the angle of the sector is 60 degrees (360 degrees divided by 6).
Substitute the values of "θ" (60 degrees) and the radius of the circle (6 inches) into the formula to find the area of one sector.
Multiply this area by 6 to get the total area of all six sectors formed by the equilateral triangle.
3. Find the area of the chord:
For an equilateral triangle inscribed in a circle, each side of the triangle is also a chord of the circle.
The area of the segment cut off by one side of the triangle is equal to the area of the sector minus the area of the equilateral triangle.
4. Subtract the area of the equilateral triangle from the total area of the sector to find the area of the segment.
Remember to use the appropriate units for the calculations.
What is the area of the circle, minus the area of the traiangle? Now divide that by three.
Think on that.