so this video is an introduction into circles the radius of a circle is a segment that connects the center of the circle which we'll call a to any point on a circle so segment a b represents the radius of a circle the diameter passes through the center of the circle and is twice the length of the radius so this is the diameter and let's call this a b c so segment abc is the diameter of a circle now on the left the center of the circle is point a so you could describe the circle like this on the right you can describe the circle based on the center point b now the area of a circle is pi r squared and the circumference of a circle is two pi r the circumference of a circle is basically the perimeter of the circle is the distance around the circle now the diameter is twice the radius so we can replace 2r with the diameter so you can also calculate the circumference using this equation it's pi times the diameter now sometimes you may need to calculate only a section of the circle for example let's say this is point a b and c and let's say b is the center of the circle and we wish to calculate the area of this sector how can we do so and also how can we calculate the length of this arc which i'm going to call s so if b is the center of the circle that means segment a b and segment bc represents the radius of the circle now to calculate the shaded region the area of the shaded region rather it's very similar to the area of a circle but with some adjustments notice that we have a fraction of the circle so whatever this angle is theta in degrees it's going to be theta divided by 360 because the entire circle has an angle of 360. so theta over 360 represents the fraction of the circle multiplied by the area of the circle so this equation will give you the area of a sector or the shaded region that we have there now to calculate the arc length of that sector it's basically a fraction of the circumference so it's theta divided by 360 times the circumference of the circle so that's how you can calculate the arc length so s the arc between a and b you can describe it this way so this represents the arc between points a and point c and the measure of arc a to c that's an angle so in this example it's theta so theta represents the measure of arc ac to calculate the length of arc ac you can use this formula now let's talk about chords let me draw a better circle it's not really that much better now what exactly is a chord a chord is a line segment that connects two points that are on the edge of the circle so a b is a chord cd is also a chord now let's say that this point is the center let's call this d e f so we have circle e or with center e and df is a chord now a chord that passes through the center of a circle is also known as the diameter of the circle so df is basically a chord and it's the diameter of the circle which means e f represents the radius of the circle and d e is also the radius of a circle so a chord is simply a line segment that starts at one end of the circle and touches another point on the circle now what happens if we form an angle using two chords so the two chords are chord a b and chord bc and notice that this angle angle abc which is known as an inscribed angle it has an intercepted arc ac which we can write it like this now let's say if the inscribed angle is 50 degrees so if angle abc is equal to 50 degrees what do you think the measure of arc ac is it turns out that the intercepted arc has an angle that's twice the value of the inscribe angle so this is going to be 100 degrees so make sure you understand that now let's say if you have an angle that touches the center of the circle as opposed to a point on a circle so let's say c is the center and this is 40 degrees and let's call this point a and point b what is the measure of arc a b the measure of arc a b is the same as the angle that touches the center of the circle so it's going to be 40 degrees but if you have an inscribed angle let's say this is 80 the inscribed angle is half of the intercepted arc so this is going to be 40. now let's work on some problems so we're given the diameter of a circle and it's eight centimeters our goal is to calculate the circumference and the area of the circle so let's draw a picture so this is the diameter this is eight centimeters the circumference is simply pi times the diameter so it's pi times 8 or you can write it as 8 pi so that's the exact answer of the circumference now if you want to get the decimal value for that just type in a pi in your calculator and so you should get 25.1 centimeters as a circumference so that's a rounded answer now the area is pi r squared so first we need to calculate the radius the radius is one half of the diameter so it's half of eight which is four centimeters so the area is going to be pi times four squared or 16 pi so 16 pi is the exact answer and the units for that are square centimeters this is in centimeters and 16 pi is 50.27 square centimeters as a decimal approximately number two the area of a circle is 81 pi what is the circumference of the circle so we know the area is pi r squared and the area is 81 pi so let's calculate the value of r so first we could divide both sides by pi if we do so these will cancel so 81 is equal to r squared and now we just got to take the square root of both sides the square root of 81 is 9. and so that's the radius it's 9 units long the circumference is 2 pi r so it's 2 pi times 9 which is 18 pi units so as a decimal that's 56.55 units long now the diameter is twice the value of the radius so it's 2 times 9 or 18. and so that's the length of the diameter of a circle number three the radius of the circle below is seven inches what is the angle measure of arc ac so this is arc ac so the angle measure of arc ac is equal to this angle where b is the center of the circle so bc is seven inches long and a b is seven inches long so the measure of the arc is 150. now what is the length of arc ac so to calculate the arc length which you can denote ss it's equal to theta divided by 360 times the circumference of a circle which is two pi r so this angle is 150 we need to divide that by 360. and then multiply that by two pi times the radius which is seven so 150 divided by 360 times two times seven this turns out to be 35 pi over six now if you want to see how to get that answer without a specialized calculator first we can cancel the zero and 15 we can reduce that to five times three thirty six that's three times twelve and twelve i'm going to break that into six times two and we have two and seven on the outside so notice i can cancel a three and i can cancel a two so i'm left over with five times seven which gives me 35 and i have a six on the bottom so that's where the six come from and plus we have pi so the arc length so arc ac has a length of 35 pi over six now as a decimal this is 18.3 now let's calculate the area of the shaded region so the area of the blue shaded region is going to be the angle divided by 360 times the area of the entire circle so the angle is 150 and we're going to divide that by 360 and the radius is 7 so we're going to multiply it by 7 squared so once again we could cancel 0 15 is 5 times 3 36 is 12 times 3. and seven squared is forty nine so forty nine times five is two forty five so this is going to be 245 pi divided by twelve so that's the exact answer and as a decimal this is about 64.1 square units so this represents the area of the sector calculate the value of x and y in each figure shown below so for the circle on the top left we can see that b is the center of the circle therefore x and the measure of arc ac has to be the same so for the first problem x is equal to 80 degrees now for the second problem on the top right the inscribe angle is going to be half of the intercepted arc so y is one half of sixty so in this case it's 30 degrees now for this one the intercepted arc is twice the value of the inscribed angle so x is going to be twice the value of 80. so it's 2 times 80 or 160 degrees now what about the third problem on the bottom left one thing i forgot to do is tell you what we're trying to calculate so let me do that now what is the value of x in this figure where x represents this angle right here so notice that it intercepts these two points which we can call bc and notice that a is the center of the circle so therefore chord bc is the diameter which bisects the circle into two parts therefore half of 360 is 180 so the intercepted arc is 180 which means that the inscribed angle is half of the intercepted arc so x is going to be half of 180 which means that it's 90. so this triangle is a right triangle number five if a b is 48 and bc is 14 what is the area of the shaded region so the area of the shader region is the difference as would represent the area of the shear region it's the difference between the area of the circle ac minus the area of the triangle a t now the area of a circle is pi r squared the area of a triangle particularly a right triangle it's one-half base times height so triangle abc which looks like this i'm going to draw it this way so this is the hypotenuse so this is vertex b this is a and this is c now a b is 48 and bc is 14. now notice that ac is the equivalent of the diameter of the circle if we can calculate the diameter we can calculate the radius now bc is the base of the triangle so bc is 14. so lowercase b in this equation is 14. the height of the triangle is 48. that's a b so now we just got to calculate the radius so let's focus on calculating the diameter first so for a right triangle we can use the pythagorean theorem c squared is equal to a squared plus b squared c is the hypotenuse which is across the box so that's the same as the diameter a is going to be 14 you can make a or b14 so if a is 14 b has to be 48. now 14 squared is 196. 48 times 48 is 2304 so if we add up 2304 plus 196 that will give us 2500 and that's equal to d squared so to calculate d we got to take the square root of both sides 2500 is the same as 25 times 100 the square root of 25 is 5 and the square root of a hundred is ten so five times ten is fifty so the diameter is fifty the radius is one half of the diameter so it's half of fifty which is twenty five so now we have everything that we need to calculate the area of the shaded region so it's going to be pi times 25 squared so 25 squared is 625 half of 14 is 7 and 7 times 48 is 336. so this is the exact answer it's 625 pi minus 336 now let's get the answer as a decimal 625 pi is 1963.5 if we subtracted by 336 this is equal to 627.5 and so this is the area of the shaded region you