Triangle and circle length and areas

In summary, the circular sector with radius 4 cm and subtending an angle of 0.8 radians has an area of 6.4 cm^2.
  • #1
karush
Gold Member
MHB
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the following diagram shows a circle with center \(\displaystyle O\) and a radius
\(\displaystyle 4cm\)
View attachment 1006
The points \(\displaystyle A, B,\) and \(\displaystyle C\) Lie on the circle.
The point \(\displaystyle D\) is outside the circle, on \(\displaystyle (OC)\)
Angle \(\displaystyle ADC=0.3\) radians and angle \(\displaystyle AOC=0.8\) radians

(a) find \(\displaystyle AD\)

I used law of sines

\(\displaystyle \frac{4}{\sin{0.3}}=\frac{x}{\sin{0.8}}\)
\(\displaystyle x \approx 9.71cm\)

there are more questions to this but want to make sure this is correct:cool:
 
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  • #2
Re: triangle and circle length and areas

Yes, you have correctly applied the Law of Sines. (Sun)
 
  • #3
Re: triangle and circle length and areas

(b) find OD

since \(\displaystyle \angle{DAO}\) is not given the its radian measure is
\(\displaystyle 2-0.3-0.8=0.9\)

so using law of sines again

\(\displaystyle \frac{x}{\sin(0.9)}=\frac{4}{\sin(0.3)}\)

so \(\displaystyle x \approx 10.6\)

 
  • #4
Re: triangle and circle length and areas

The sum of the interior angles of a triangle in radians is $\pi$, not $2$.

Also, when you go to use this, be aware of the identity $\sin(\pi-\theta)=\sin(\theta)$.
 
  • #5
Re: triangle and circle length and areas

\(\displaystyle \pi-1.1 \approx 2.04159\)

\(\displaystyle \frac{x}{sin(2.4159)}=\frac{4}{sin(0.3)}\)

\(\displaystyle x \approx 13.53\)
 
  • #6
Re: triangle and circle length and areas

You appear to have dropped a zero to the right of the decimal point in the argument for the sine function on the left. I would write:

\(\displaystyle \frac{\overline{OD}}{\sin(A)}=\frac{4\text{ cm}}{\sin(0.3)}\)

Now, given:

\(\displaystyle 0.3+0.8+A=\pi\,\therefore\,A=\pi-1.1\)

and using the identity \(\displaystyle \sin(\pi-\theta)=\sin(\theta)\)

we have:

\(\displaystyle \frac{\overline{OD}}{\sin(1.1)}=\frac{4\text{ cm}}{\sin(0.3)}\)

\(\displaystyle \overline{OD}=\frac{(4\text{ cm})\sin(1.1)}{\sin(0.3)}\approx12.062895734\text{ cm}\)
 
  • #7
View attachment 1009

(c) find the area of sector \(\displaystyle OABC\)

\(\displaystyle \Bigg(\frac{0.8}{2\pi}\Bigg)\Bigg(\pi 4^2 \Bigg)\approx 6.4 cm^2\)

last question

(d) Find the area of region \(\displaystyle ABCD\)

\(\displaystyle \frac{1}{2} (12.0624)(4\sin{0.8})-6.4 \approx 10.91 cm^2\)
 
  • #8
c) The area $A$ of a circular sector having radius $r$ and subtending an angle $\theta$ is given by:

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

In this case what are $r$ and $\theta$?
 
  • #9
MarkFL said:
c) The area $A$ of a circular sector having radius $r$ and subtending an angle $\theta$ is given by:

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

In this case what are $r$ and $\theta$?

r=4 and \theta = 0.8

so

\(\displaystyle A=\frac{1}{2} 4^2 \ (0.8)=6.4 cm\)
 
  • #10
The magnitude of the result is correct, but the unit of area is not, which may seem very minor now, but if you take physics, keeping track of the units becomes important, and it is a good habit to get into early on. I would write:

\(\displaystyle A=\frac{1}{2}(4\text{ cm})^2\cdot0.8=6.4\text{ cm}^2\)

You should expect an area to have as its unit of measure the square of a linear measure.
 
  • #11
looks like just need to be more careful:)

thanks again for help
 

FAQ: Triangle and circle length and areas

What are the formulas for finding the perimeter and area of a triangle?

The formula for finding the perimeter of a triangle is simply adding the length of each side. The formula for finding the area of a triangle is 1/2 * base * height.

What is the Pythagorean theorem and how is it used to find the length of a side of a triangle?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem can be used to find the length of a side by rearranging the formula to solve for the missing side.

How do I find the circumference and area of a circle?

The formula for finding the circumference of a circle is 2 * pi * radius. The formula for finding the area of a circle is pi * radius^2. Remember to use the same unit of measurement for both the radius and the circumference or area.

Can I use the same formula to find the length of a curved side of a triangle as I would for finding the circumference of a circle?

No, the formula for finding the circumference of a circle only applies to circles. To find the length of a curved side of a triangle, you would need to use the Pythagorean theorem or a trigonometric function.

What is the relationship between the area and circumference of a circle?

The area of a circle is directly proportional to the square of its circumference. This means that if you double the circumference of a circle, its area will be four times as large. This relationship can also be seen in the formulas for finding the circumference and area of a circle, where the radius is squared in the area formula but not in the circumference formula.

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