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11.4 & 11.5 Notes

Added Apr 23, 2015 •

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11 for an 11-5 circles now this is just area and circumference of circles which you should know the basics for that are ready but definition of circumference the distance around a circle now that's basically the same as perimeter however they have a new name for it when you talk about a circle in the biggest difference is because tell a formula for circumference and you do not need a formula for perimeter so that's how their distinguished just a little bit but they're both the distance around the outside circumference best for a circle so your formula is circumference equals 2 pi r or you can do just pie what's the diameter because think about it two times the radius is just the diameter so you can place the two are with a deep so you're going to use one of those too remember this is just like perimeter so it's going to be measured in single units  feet inches inches centimeters and on the first two examples that we have right here number one we have a radius of 4 Raiders so if they give us the radius I would use this one if you'll know soon number to they give us a diameter I would use this one either way is fine for you to so I'll look at number one and let's set up the problem so if we use C equals 2 pi r that would just be too pi times the radius is 4 top 10 in your calculator and when you do that make sure you use the pie button on your calculator it's above the little house top under the are buttoned it's more accurate than 3.14 because Pi is a really long number and the more you round it the more inaccurate you're going to be I want you put that in your calculator you get 25.13 and then we're in meters so to be 25.13 meters around that Circle that's how far it is if you have a radius of 4 no look at number two number two has the diameter so I would use Pi * D so that would be pie times 12.5 not remember use the pie button on your calculator x 12.5 and you get 39-point U7 is what it rounds to in that is in feet so this circle with a diameter of 12 and a half feet would have 39.27 ft around outside all right now look at number 3 a tire has a diameter of 26 in how far does the tire travel when it makes 50 revolutions now this is really how it car calculate the mileage or your odometer in your car is it how many times your tires rotate there's a sensor Nair and it knows the size of your tires and how far it goes every time you're tired makes one rotation so if we do that  since it gives us an ammeter I would use C equals Pi D soap find a circumference of that tire so pi x 26 81.68 now that's one rotation of a tire cuz that's how big the tire is next 50 revolutions you would multiply that by 50 and that car has traveled 4084 in now what would that be feet just convert / 12 that's the same as 340 ft that that car has gone in 50 revolutions okay the other thing we want to talk about his arklink now what if we don't want to know the distance around the whole circle we just want to know a portion of that Circle will that would be an arc length so if you have a circle remember an angle cuts off an art that's your art what if we just need to know that distance of that that's the Arc Length know what we were doing in the past was this Eric was equal to its central angle now that was a degree measure we won actual distance here so let's find the distance that is and it has something to do with that angle again but are clean is an pulling is portion of the circumference of the circle came the measure is in degrees the arc length is a distance measure so we're trying to find a distance not the degree measure with the degree measure that was equal to the angle measure for your central angle alright so we have a formula there's a different formula I think when is the easiest just make a ratio proportion if the arc is a proportion of the circumference it's this part of the whole it's equal to this angle out of Hall 360 so it's the same portion of the circle that that angle is of the inside of the circle so if you do the arc measure that's the measure / 360 that's the angle measure I mean equals The Arc Length out of the so  this link that we want goes here the whole circumference goes here the arc measure which is equal to the central angle out of three now look at these two examples the first thing is do not forget that when we are trying to find an arc measure like a Define arc length of a beat it has to have three letters to name the major arc so we're just looking for this Ark right here what is the distance now we're going to set up that ratio proportion however we need the circumference to set that up so before we can solve anything we need to circumference to radius here is too so if we do 2 times pi times the radius of 2 you were going to get 12.6 or your circumference is 12.6 now you set up and you do the angle  how to 360 equals the arc measures what we want X Out of the entire circumference so we won't this portion out of the circumference tell me what this angle out of 360 so you just cross multiply and when you do that you will get 360 equals  1000 108.8 so divide each side by 360 and you're going to get that x equals 3.08 so that Arc right here is three point 8 meters out of your entire circumference 12.6 meters knots meters right there okay so that what is 3.08 M out of 12.6 M 5 in let's look at that again we have a radius and we need to find the circumference so two-part sew 2 * pi * 12.3 inner circumference is going to be 7 7.3 ft okay now we don't have this angle this is find the measure of a deep we have the link the vaidy it's 38 but we need to find the actual measure your ex is going to be in a different spot so remember it's the arc measure out of 360 or the angle out of 360 The Arc Length out of the entire circumference so do that cross multiply you get 77.3 X schools  13680 / 77.3 and you get the x equals 176 .97  feet  so that is the actual angle so that would be this angle measure right hear 176.97 degrees on this Arc would be 176.97 degrees okay that's circumference and Arc Length now let's talk about area and area is the amount of space that is inside of this circle blue have 20.5 CM right here for this now the formula for area is pi times the ray squared pi r squared that is not the same as circumference it's two pie are so if I'm going to find the area of this circle area equals I tons of radius squared so it's pie times now this gives me the diameter in this case when you're fun and area you have to have the radius you're going to do half of 20.5 and 1/2 of 20.5 is 10.25 + square not remember you only Square the you do not Square the pie so you must Square the radius before you X the pot now if you type this in your calculator just exactly like it looks what there then you will get the right answer cuz your calculator will do your order of operations for you but if you type that in your going to get an area of 330.06 now this is area so it's centimeters squared so there's 330.06 cm squared in that circle now just like we wanted to find an arc length apart of the circumference we want to find a sector of the Circle K which is just a portion it's not the entire area inside it's like you want to find the area of one piece of pie instead of the whole pie so sector of a circle is a region Bound by two radii  have a circle and their intercepted Arc so it's this area right here that's made by this radius in this radius and the intercepted Arc so it's that piece of the pie so to say so it's a some of the areas in we're going to make another ratio proportion because it'll be this portion out of the whole area equals this angle right here the measure of that angle or this art either one cuz they're equal out of 360 which is the total measurement of the whole circle so here's your formula Arc measure or the angle measure out of 360 equals the area of the sector out of the whole area of the circle okay so look at example number 7 it says fun that Yuri the sector's formed by angle in BV this one right here in be alright so this angle we know right here is 68 so this Arc measure right here is 68 degrees now can you on the major arc on this other side right here it's degrees and that would be 360 - 68 so this one up here would be 290 7 degrees so we can find each one now the first thing you got to know is the area because we're talking about sectors not Arc Length so the area pi times the radius is 4 squared now remember type this in your calculator Shaklee like it looks Pi * 4 squared only Square the four that's 50.2 7 and that is in meters where we want to find the small Arc and the large one so let's look at the small one first  so it'd be this Ark in I just noticed set 63 and I label that 68 change that to 63 I'm sorry that is right 297 though so it will be this angle measure where this Arc measure out of the whole circle so out of 360 so we'll do 63 out of 360 pause the area of the sector which is what we want to find out of the entire area of the whole circle which is what we found up here so that is 50.2 okay when you cross multiply and solve for x you get that the area of that sector is 8.87 there's Square now that's the large will think about it I mean that's a small and I'm sorry the large one should be what's left over of the entire Berryessa really you should be able to do the entire area - that one should leave you all of this but let's work it out just so you can see all right so this one going to be the arc measure which is 297 so this angle is 297 out of 360 degrees equals equals the area of the sector X Out of the entire area 50.2 seven know when you cross multiply that install 4X who end up getting 41.47 M squared now if you add those together you should get 50.27 I think we rounded a little bit we may be off by .1 or so but this would be the area of the small sector that's how much area where in this will be the area of the larger sector that's how much area is here and together they make up that alright now look at number 8 parade it says use the diagram to find the area of Circle C Suite once the total area and what it gives you is the area of this sector right here so this is a 45° Google and it has an area of 22 M squared so if you set up your ratio proportion remember it's the angle out of 360 equals the area of the sector out of the total area of the circle which would be X now you just cross multiply 45 x equals 7920 divided by 45 and X is going to equal 176 M squared so there the entire Cirque is 176 M squared and the area of that one sector is 22 now you can subtract the 22 from that and find the rest of the area of that part all right now you have a worksheet to do for this on it do the evens only its 11.4 and 11.5 it will be due Tuesday and we will take a quiz

Uploaded by Pam Chastain.

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11.4 & 11.5 Notes

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