Application of Circular Measures

1.4 Application of Circular Measures
 
Mind map illustrating circular measures in construction, featuring equations for degrees, radians, arc length, and sector area.
 
Real Life Application
  • The rainbow shown in the photo is an arc of a circle. One of the application is to determine the length of the arc of the rainbow.
  • The cross-section of a train tunnel is usually in the form of a major arc of a circle. The arc length and the area of this cross-section tunnel can be calculated using the formulas for circular measure.
 
Example
Question

The diagram below shows the sectors \(OAB\) and \(OCD\) with a common center \(O\).

Diagram illustrating sector AOB and COD, both at 75 degrees, with lengths OA at 3k and BD at k.

Find

(a) the value of \(k\) if the area of the shaded region is \(13.09\) cm\(^2\),
(b) the perimeter of the shaded region.

Solution

(a)

Change the angle to radian,

\(75^\circ\times \dfrac{\pi}{180^\circ}=1.3092\text{ rad}.\)

Use the formula of area of sector,

\(\text{Area}=\dfrac{1}{2}r^2\theta\)

Area of shaded region:

\(=\text{Area of sector }OAB-\text{Area of sector }OCD\)

\(\begin{aligned} 13.09&=\dfrac{1}{2}(3k)^2(1.3092)-\dfrac{1}{2}(2k)^2(1.3092) \\\\ 13.09&=\dfrac{1}{2}(9)(1.3092)k^2-\dfrac{1}{2}(4)(1.3092)k^2 \\\\ 13.09&=5.8914k^2-2.6184k^2 \\\\ 13.09&=3.273k^2 \\\\ k&=\sqrt{\dfrac{13.09}{3.273}}\\\\ k&=1.9998\approx2\text{ cm}. \end{aligned}\)


(b)

Use the formula of arc length:

\(s=r\theta\)

Perimeter of the shaded region:

\(=k+\text{Arc }CD+k+\text{Arc }AB\)

\(\begin{aligned} &=2+2+2(2)(1.3092)+3(2)(1.3092) \\ &=4+5.2368+7.8552 \\ &=17.092\text{ cm}. \end{aligned}\)

 

Application of Circular Measures

1.4 Application of Circular Measures
 
Mind map illustrating circular measures in construction, featuring equations for degrees, radians, arc length, and sector area.
 
Real Life Application
  • The rainbow shown in the photo is an arc of a circle. One of the application is to determine the length of the arc of the rainbow.
  • The cross-section of a train tunnel is usually in the form of a major arc of a circle. The arc length and the area of this cross-section tunnel can be calculated using the formulas for circular measure.
 
Example
Question

The diagram below shows the sectors \(OAB\) and \(OCD\) with a common center \(O\).

Diagram illustrating sector AOB and COD, both at 75 degrees, with lengths OA at 3k and BD at k.

Find

(a) the value of \(k\) if the area of the shaded region is \(13.09\) cm\(^2\),
(b) the perimeter of the shaded region.

Solution

(a)

Change the angle to radian,

\(75^\circ\times \dfrac{\pi}{180^\circ}=1.3092\text{ rad}.\)

Use the formula of area of sector,

\(\text{Area}=\dfrac{1}{2}r^2\theta\)

Area of shaded region:

\(=\text{Area of sector }OAB-\text{Area of sector }OCD\)

\(\begin{aligned} 13.09&=\dfrac{1}{2}(3k)^2(1.3092)-\dfrac{1}{2}(2k)^2(1.3092) \\\\ 13.09&=\dfrac{1}{2}(9)(1.3092)k^2-\dfrac{1}{2}(4)(1.3092)k^2 \\\\ 13.09&=5.8914k^2-2.6184k^2 \\\\ 13.09&=3.273k^2 \\\\ k&=\sqrt{\dfrac{13.09}{3.273}}\\\\ k&=1.9998\approx2\text{ cm}. \end{aligned}\)


(b)

Use the formula of arc length:

\(s=r\theta\)

Perimeter of the shaded region:

\(=k+\text{Arc }CD+k+\text{Arc }AB\)

\(\begin{aligned} &=2+2+2(2)(1.3092)+3(2)(1.3092) \\ &=4+5.2368+7.8552 \\ &=17.092\text{ cm}. \end{aligned}\)