The central angle of two sectors

Answer

Verified

Hint: To solve this question one should need to know the concept of radius and arc concerning the angle theta ($\theta $). The basic formulas are required to solve this question.We must use these formulas$Area(\operatorname{Sec} tor) = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi \times {r^2}$ --(1) where $\theta $is in degrees and $r$=radius in given units.Arc Length of Sector = $r \times \theta $ --(2)where $\theta $is in radians and $r$=radius in given units.

Complete step-by-step answer:

Step1: Collect all the data or information given in the question. The given information is: Radius of circle 1(${r_1}$) = 7cmRadius of circle 2(${r_2}$) = 21cmCentral angles of circle 1(${\theta _1}$) = ${120^0}$ Central angles of circle 2(${\theta _2}$) = ${40^0}$ Step2: Now we will understand the demand of the question which is to find the areas of a sector as well as the length of arcs of two circles.Step3: The area formula we will use from the hint section which is given by equation (1).$ \Rightarrow Area(\operatorname{Sec} tor) = \dfrac{\theta }{{{{360}^ \circ }}} \times \pi \times {r^2}$ Now we will find the area for the first circle,\[ \Rightarrow {(Area)_1} = \dfrac{{{{120}^0}}}{{{{360}^0}}} \times \pi {(7)^2}\] \[\therefore {(Area)_1} = \dfrac{{49\pi }}{3}c{m^2}\]Similarly, we will find the area for the second circle,\[ \Rightarrow {(Area)_2} = \dfrac{{{{40}^0}}}{{{{360}^0}}} \times \pi {(21)^2}\]\[ \Rightarrow {(Area)_2}\]$ = \dfrac{1}{9} \times \pi (441)$ \[\therefore {(Area)_2} = 49\pi c{m^2}\]\[\] Step4: Now we will find the length of the two arcs.\[ \Rightarrow Angle = \dfrac{{arclength}}{{radius}}\] Where angle used in the formula needs to be in radian but the given angle in question is in degrees so we will first try to convert that to radians by using equation (3).Angle for the first circle, $ \Rightarrow {120^0} = \left( {\dfrac{\pi }{{180^\circ }} \times 120^\circ } \right)radians$ $\therefore {\theta _1} = \left( {\dfrac{{2\pi }}{3}} \right)radians$Angle for the second circle, $ \Rightarrow {40^0} = \left( {\dfrac{\pi }{{180^\circ }} \times 40^\circ } \right)radians$ $\therefore {\theta _2} = \left( {\dfrac{{2\pi }}{9}} \right)radians$ Arc length of circle 1$({S_1}) = \dfrac{{2\pi }}{3} \times radiu{s_1}$ $ \Rightarrow $Arc length of circle 1$({S_1})$$ = \dfrac{{2\pi }}{3} \times 7$ $\therefore $Arc length of circle 1$({S_1})$$ = \dfrac{{14\pi }}{3}cm$Arc length of circle 2$({S_2}) = \dfrac{{2\pi }}{9} \times radiu{s_2}$$ \Rightarrow $Arc length of circle 2$({S_2}) = \dfrac{{2\pi }}{9} \times 21$$\therefore $Arc length of circle 2$({S_2})$$ = \dfrac{{14\pi }}{3}cm$ Final Answer: The arcs of circle 1 and circle 2 are \[\dfrac{{49\pi }}{3}c{m^2}\] and $49\pi c{m^2}$ respectively. Similarly, the value of arc lengths for circle 1 and circle 2 is both $ = \dfrac{{14\pi }}{3}cm$.

Note: This question is pretty simple and based on the concept of circular arcs. So to solve it successfully students should be very clear about the basic numerical formulas and related terminologies of arc, such as arc length, area of arc, etc. Apart from this students should know about the conversion of angle from degrees to radians.

Formula to convert in degrees to radians and vice-versa.$\pi (radians) = {180^0}(\deg rees)$

Updated November 03, 2020

By Mariecor Agravante

Circles are everywhere in the real world, which is why their radii, diameters and circumference are significant in real life applications. But there are other parts of circles – sectors and angles, for instance – that also have importance in everyday applications as well. Examples include sector sizes of circular food like cakes and pies, the angle traveled in a Ferris wheel, the sizing of a tire to a particular vehicle and especially the sizing of a ring for an engagement or wedding. For these reasons and more, geometry also has equations and problem calculations dealing with central angles, arcs and sectors of a circle.

The central angle is defined as the angle created by two rays or radii radiating from the center of a circle, with the circle’s center being the vertex of the central angle. Central angles are particularly relevant when it comes to evenly dividing up pizza, or any other circular-based food, among a set number of people. Say there are five people at a soiree where a large pizza and a large cake are to be shared. What is the angle that both the pizza and the cake have to be divided at to ensure an equal slice for everyone? Since there are 360 degrees in a circle, the calculation becomes 360 degrees divided by 5 to arrive at 72 degrees, so that each slice, whether of the pizza or the cake, will have a central angle, or theta (θ), measuring 72 degrees.

An arc of the circle refers to a “portion” of the circle’s circumference. The arc length therefore is the length of that “portion.” If you imagine a pizza slice, the sector area can be visualized as the entire slice of pizza, but the arc length is the length of the outer edge of the crust for that particular slice. From the arc length, the central angle can be calculated. Indeed, one formula that can help in determining the central angle states that the arc length (s) is equal to the radius times the central angle, or

s = r \times θ

where the angle, theta, must be measured in radians. So to solve for the central angle, theta, one need only divide the arc length by the radius, or

\frac{s}{r} = θ

To illustrate, if the arc length is 5.9 and the radius is 3.5329, then the central angle becomes 1.67 radians. Another example is if the arc length is 2 and the radius is 2, the central angle becomes 1 radian. If you want to convert radians to degrees, remember that 1 radian equals 180 degrees divided by π, or 57.2958 degrees. Conversely, if an equation asks to convert degrees back into radians, then first multiply by π, and then divide by 180 degrees.

Another useful formula to determine central angle is provided by the sector area, which again can be visualized as a slice of pizza. This particular formula can be seen in two ways. The first has the central angle measured in degrees so that the sector area equals π times the radius-squared and then multiplied by the quantity of the central angle in degrees divided by 360 degrees. In other words:

πr^2 × \frac{\text{central angle in degrees}}{360 \text{ degrees}} = \text{sector area}

If the central angle is measured in radians, the formula instead becomes:

\text{sector area} = r^2 × \frac{\text{central angle in radians}}{2}

Rearranging the formulas will help to solve for the value of the central angle, or theta. Consider a sector area of 52.3 square centimeters with a radius of 10 centimeters. What would its central angle be in degrees? The calculations would begin with a sector area of 52.3 square centimeters being equal to: