Calculate the radius of the circle

In summary, the conversation discusses points A, B, and C on a circle with center O, where angle ABC is equal to 75 degrees and the area of the shaded segment is 200cm^2. The question is to calculate the radius of the circle and the angle AOC, as well as the area of the triangle AOC and the sector AOC. To find the radius, an equation is provided using the known angle measures and areas. The conversation also includes a discussion about angles and arcs on a circle and how to find the area of the gray region as a function of the radius. The final equation for the shaded segment is provided as $(\frac{5\pi}{12}- sin(75)cos(
  • #1
Help seeker
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A, B and C are points on a circle with center O. Angle ABC = $75°$ . The area of the shaded segment is $200cm^2$ .
Picture1.png

Calculate the radius of the circle. Answer correct to $3$ significant figures.
 
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  • #2
What is the angle AOC? In terms of $r$ (the radius of the circle), what is the area of the triangle AOC? What is the area of the sector AOC of the circle? The difference between those last two areas is the area of the shaded segment. That should give you an equation for $r$.
 
  • #3
An angle with vertex on a circle cuts that circle in an arc with measure twice the measure of the angle. And an angle with vertex at the center of the circle cuts that circle in an arc with measure equal to the angle. Here angle ABC has measure 75 degrees so arc AC and angle AOC have measure 150 degrees. Knowing that, you can find the areas of segment AOC and triangle AOC as functions of r, the radius of the circle, Subtract those to areas to find the area of the gray region as a function if r, set that equal to 200, and solve for r.
 
  • #4
Since this has been here a while:
Sector AOC is 150/360= 15/36= 5/12 of the entire circle, The entire circle has area $\pi r^2$ so sector AOC is $5\pi r^2/12$. Triangle AOC is an isosceles triangle with two sides of length r and vertex angle 150 degrees. An altitude of triangle AOC bisects angle AOC so divides triangle AOC into two right angles with hypotenuse of length r and one angle 150/= 75 degrees. The "near side", the altitude, has length h= r cos(75) and the "opposite side", half of AC, has length b= r sin(75). All of AC, the base of triangle AOC, has length 2r sin(75) so triangle (1/2)hb= r^2 sin(75)cos(75).

So the shaded segment has area $\frac{5\pi r^2}{12}- r^2sin(75)cos(7)= \left(\frac{5\pi r^2}{12}- sin(75)cos(75)\right)r^2$.
 
  • #5
Tnx.
SOLVED
 
  • #6
Brandy induced ramblings follow.
Country Boy said:
...
So the shaded segment has area \(\displaystyle \frac{5\pi r^2}{12}- r^2sin(75)cos(7)= \left(\frac{5\pi r^2}{12}- sin(75)cos(75)\right)r^2\)
Minor typo.
That should have been
\(\displaystyle \frac{5\pi r^2}{12}- r^2sin(75)cos(75)= \left(\frac{5\pi}{12}- sin(75)cos(75)\right)r^2\)
 

Related to Calculate the radius of the circle

1. How do you calculate the radius of a circle?

The radius of a circle can be calculated by dividing the diameter of the circle by 2. Alternatively, you can use the formula: radius = circumference / (2 * pi).

2. What is the formula for calculating the radius of a circle?

The formula for calculating the radius of a circle is: radius = diameter / 2 or radius = circumference / (2 * pi).

3. Can you use the circumference to calculate the radius of a circle?

Yes, you can use the circumference to calculate the radius of a circle by dividing the circumference by 2 * pi.

4. How do you measure the diameter of a circle?

The diameter of a circle can be measured by drawing a straight line passing through the center of the circle and measuring the length of the line.

5. What units are used to measure the radius of a circle?

The radius of a circle is typically measured in units of length, such as centimeters, meters, or inches.

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