Find the areas of segment in circle

In summary: DIn summary, we can use the formula for the area of a sector (A = 1/2 * r^2 * theta) and the area of a triangle (A = 1/2 * base * height) to find the area of shared regions. By calculating the area of the sector and subtracting the area of the triangle, we can determine the shaded area. Similarly, we can find the sum of the area of the sector and the area of the triangle to get the total area of the shared region.
  • #1
Etrujillo
9
0
So far i have.
14) area of sector is πr²/3 = 12π
length of chord. that triangle has two sides of 6 and angle of 120º
split the triangle in two right triangles with angle of 120/2 = 60 and hyp=6. other (longer) side is:
sin 60 = x/6
s = 6 sin 60 = 6(√3/2) = 3√3
third side is
s = 3 cos 60 = 3
area =(1/2)(3)(3√3) = 4.5√3, double for both triangles
subtract that from the sector to get (12π) – (9√3)

15) similar to above.
find the area of the 270º sector and add the area of the triangle
area of sector is (270/360)(π9²) or (3/4)81π
area of triangle is (1/2)81

Is this correct?
What can i do differently?

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  • #2
Re: Find the area of the shared region.

14.) I would take the area of the circular sector, and subtract from it the area of the triangle to get the shaded area \(A\):

\(\displaystyle A=\frac{1}{2}r^2\theta-\frac{1}{2}r^2\sin(\theta)=\frac{r^2}{2}(\theta-\sin(\theta))\)

Next, we identify:

\(\displaystyle r=6\text{ cm}\)

\(\displaystyle \theta=\frac{2\pi}{3}\)

Hence:

\(\displaystyle A=\frac{(6\text{ cm})^2}{2}\left(\frac{2\pi}{3}-\sin\left(\frac{2\pi}{3}\right)\right)=3\left(4\pi-3\sqrt{3}\right)\text{ cm}^2\quad\checkmark\)

This is equivalent to the area you stated. :D

15.) I would find the sum of 3/4 of the area of the circle and the right isosceles triangle:

\(\displaystyle A=\frac{3}{4}\pi r^2+\frac{1}{2}r^2=\frac{r^2}{4}(3\pi+2)\)

We identify:

\(\displaystyle r=9\text{ in}\)

And so:

\(\displaystyle A=\frac{(9\text{ in})^2}{4}(3\pi+2)=\frac{81}{4}(3\pi+2)\text{ in}^2\quad\checkmark\)

This is equivalent to what you would get when you add the two areas you found.
 

FAQ: Find the areas of segment in circle

What is a segment in a circle?

A segment in a circle is a region bounded by a chord and an arc of the circle. It is a portion of the circle that is cut off by a chord.

How do you find the area of a segment in a circle?

To find the area of a segment in a circle, you need to know the central angle of the segment and the radius of the circle. Then you can use the formula A = (θ/360)πr² to calculate the area, where θ is the central angle and r is the radius.

What is the central angle of a segment in a circle?

The central angle of a segment in a circle is the angle formed by the two radii of the circle that intersect at the endpoints of the segment. It is also known as the arc angle.

Can a segment in a circle have a negative area?

No, a segment in a circle cannot have a negative area. The area of a segment is always a positive value, as it represents the amount of space within the boundaries of the segment.

What are the units of measurement for the area of a segment in a circle?

The units of measurement for the area of a segment in a circle can vary depending on the units used for the radius. For example, if the radius is given in meters, the area will be in square meters. If the radius is given in inches, the area will be in square inches.

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