Question

Express answer in exact form. Show all work for full credit. A segment of a circle has a 120 arc and a chord of 8in. Find the area of the segment.

Answer 1

Answer:

21 & 1/3 pi in.^2 - 16 sqrt 3 in.^2

Step-by-step explanation:

First, make a triangle with the chord by drawing two other lines. Each line should have an endpoint of the chord and the circle's center point. This, plus the segment, will give you a sector.

Next, bisect the chord in half with a line that has one point as the midpoint of the chord and the other point as the bisector of the central angle. This makes two 30-60-90 right triangles. This also means that now you can find out the radius of the circle by using the rules for 30-60-90 right triangles.  

You already know the long leg of the right triangle: 4 sqrt 3. From this, you can figure out that the short leg is 4 and the hypotenuse is 8.  

Now you can find out the area of the whole circle by using the formula for the area of a circle: A = pi * r^2 ----> A = 64 pi.  

Then, to find the area of the sector, you take 120o (o = degree sign) and divide it by 360o. So --> 120/360. You should get 1/3. Multiply 1/3 by the area of the circle (in this case 64 pi). So you would get 21 & 1/3 pi for the area of the sector.  

Now you need to find the area of the triangle. Use the formula for the area of triangles: A = 1/2 * b * h. We already know the base and height: back to the 30-60-90 right triangle ---> the short leg, or the height, is 4. The base was given: 8 sqrt 3. So, plug these numbers into the formula for triangles  ---> A = 1/2 * 8 sqrt 3 * 4  ---> A = 2 * 8 sqrt 3  ---> A = 16 sqrt 3.  

Finally, to find out the area of the segment, you would need to do the area of the sector minus the area of the triangle:

21 & 1/3 pi in.^2 - 16 sqrt 3 in.^2

Hope this helps!  :)

Answer 2

Check the picture below.

bear in mind that, arcs get their angle measurement, from the central angle they're in.