welcome back mathematicians in this video we are going to look at the area of a sector of the circle linear speed and angular speed let's start with the area the area a of a sector of a circle of radius r formed by a central angle of theta radians is the area equals one-half times the radius squared times theta the angle in radians central angle in terms of radians so the key here is that theta needs to be in terms of radians if at any time it's in terms of degrees to use this formula you will have to convert it over to radians so to give you a visual of what's happening if i have a circle and i draw a central angle in that circle what i need in order to calculate the area of this sector of the circle is the radius the radius being the distance from the center of the circle to where the ray intersects the circle and i need the central angle theta and theta would be in terms of radians so i have to find in my first problem i'm going to find the area of the sector of a circle of radius 5 feet formed by an angle of 40 degrees and i'm going to round the solution or answer to the nearest two decimal places let's start by converting 40 degrees to radians i'm going to do that by multiplying by pi over 180 degrees which will give me if i do the product will give me 40 pi over 180 radians and then i need to reduce and when i reduce i get 2 pi over 9 radians i am now prepared to use the formula because i have the angle in terms of radians the formula is equal to or the formula is area is equal to one-half times the radius which in this case is 5 quantity squared times the angle in terms of radians so that's 2 pi over 9. now i do encourage you to use your pi button on your calculator not 3.14 but what you get rounded to two decimal places should be 8.73 now the units are going to be feet squared squared because it's area and feet because that is the units that's the unit of the radius let's now look at linear speed and angular speed before i read this i just want to relate this back to something you are probably very familiar with the formula distance is equal to rate times time so in this case instead of a distance being a straight line distance we are going to look at the distance around a circle and because it's the distance around a circle we are going to look at the arc length so if we start right here on our arc or our circle and then we end right here we would want to know that arc length which we know is represented by the letter or variable s we then are going to look at um instead of using the r for rate we're going to instead call it v for velocity and then finally we still use t as a unit of time we are now going to solve this for v and so we get v is equal to s the arc length over a unit of time and so that is the formula that we're going to use to find linear speed now read it so it says suppose that an object moves around a circle of radius r at an at a constant speed if s is the distance traveled in time t around the circle then the linear speed v of the object is defined by v is equal to s over t now let's go to angular speed instead of speed being measured as a distance over a unit of time this time it's going to be measured over radians or an angle over unit of time so theta is an angle in radians over a unit of time which can be represented as you know hours minutes seconds etc so the angular speed omega that is omega by the way i'm not very good at writing it so it looks more like a w when i write it of an object is the angle theta measured in radians swept out divided by the elapsed time t that is omega is equal to theta over t we are now going to use that information to solve this problem where we have an object traveling around a circle with a radius of 5 centimeters if in 20 seconds a central angle of 1 3 radian is swept out what is the angular speed of the object and what is the linear speed of the object let's start with the angular speed because believe it or not we're given the information to directly find angular speed without a lot of extra work so angular speed which is omega is equal to theta that central angle in terms of radians divided by t the unit of time in this case the central angle in terms of radians is 1 3 divided by t which is 20 seconds so when i take this and simplify it what i end up getting is one sixtieth of a radian per one second okay so one sixtieth of a radian per one second is my speed now if i wanted to calculate this in terms of a decimal rounded to let's say two decimal places what i would end up getting is 0.02 radians per second okay so perfect in the perfect form it's going to be 1 60 of a radian per second and decimal would be 0.02 radians per second now in order to find the linear speed i actually need to find the arc length okay so the arc length s is equal to theta times r in this case theta again is one third and r is actually five centimeters so i'm i find the product and i get five thirds and we are dealing with centimeters so centimeters so that is my arc length now i'm going to use the linear speed formula which is v is equal to five thirds which is the arc length over unit of time which is 20 seconds when i simplify this ratio what i end up getting is 1 12 of a centimeter per second now if i go ahead and calculate the decimal rounded to two decimal places what i get is .08 centimeters per second so again linear speed is going to be a distance per unit of time so it's 5 3 centimeters for 20 seconds i go ahead and divide those and i get 1 12 of a centimeter per second and then i can divide that again 1 12 in order to get .08 centimeters per second please read the directions very carefully and understand if the directions want you to give the exact answer which would be 1 12 centimeter per second or 1 60th radian per second or if they're okay with a decimal rounded to the nearest one or two or three decimal places all right guys good luck