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An arc of a circle that is 6Cm in length intercepts a central angle of 1.5 radians find the number of Cm in the radius of the circle?
An arc of length 6cm subtending an angle at the centre of 1.5c is equivalent to the whole circle of length 2πr subtending 2π radians. Therefore,
6/1.5 = 2πr/2π = r : Then r = 4 cm.
NOTE : A radian can be defined as the angle at the centre of a circle subtended by an arc equal in length to the radius. So an arc subtending an angle of 2 radians is twice the length of the radius. An arc subtending an angle of 1.5 radians is thus 11/2 times as long as the radius.
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How do you work out the circumference of a circle using radius?
The formula for calculating the circumference of a circle is 2πr, where r is the radius of the circle and π is 3.1415926535890793 - usually shorted to either 3.1416 or 3.14 So that the circumference of a circle with a radius of 10 units is 62.83 units There are pi radians in a half of a circle. Thus, the measure of a central angle which is a straight line is pi radians. We have a formula that show that the length of an intercepted arc is equal to the product of the angle in radians that intercepts that arc, with the length of the radius of the circle. So we can say that the length of a semicircle is (pi)(r). In a full circle are 2pi radians. So the length of intercepted arc from a central angle with measure 2pi is 2(pi)(r).
A central angle of a circle of radius 30 cm intercepts an arc of 6 cm Express the central angle in radians and in degrees?
A central angle is measured by its intercepted arc. Let's denote the length of the intercepted arc with s, and the length of the radius r. So, s = 6 cm and r = 30 cm. When a central angle intercepts an arc whose length measure equals the length measure of the radius of the circle, this central angle has a measure 1 radian. To find the angle in our problem we use the following relationship: measure of an angle in radians = (length of the intercepted arc)/(length of the radius) measure of our angle = s/r = 6/30 = 1/5 radians. Now, we need to convert this measure angle in radians to degrees. Since pi radians = 180 degrees, then 1 radians = 180/pi degrees, so: 1/5 radians = (1/5)(180/pi) degrees = 36/pi degrees, or approximate to 11.5 degrees.
What is the central angle of a circle with a circumference of 9 and arc length of 1?
2pi/9 radians or 40 degrees
Find Length of an arc of circle when measure of arc is 50 and radius is 3.5?
We have a formula of finding the arc length, s = θr, where s is the length of the intercepted arc, θ is the central angle measured in radians, and r is the radius of the circle. So that we need to convert 50 degrees in radians. 1 degrees = pi/180 radians 50 degrees = 50(pi/180) radians = 5pi/18 radians s = θr (replace θ with 5pi/18, and r with 3.5) s = (5pi/18)(3.5) = (17.5/18) pi ≈ 3 Thus, the length of the arc is about 3.
How do you translate radian measure into measurements?
When you work with triangles you are using degrees to measure its angles, but when you are working with circles is more convenient to use the radian measure.What is a radian?Definition: A radian is the measure of an angle that, when drawn as a central angle of a circle, intercepts an arc whose length is equal to the length of a radius of the circle.For example,1. When a central angle of a circle with a radius 1cm intercepts an arc of the circle which is also 1cm, this angle has 1 radian measure.2. If you'll make this central angle bigger in order to intercept an arc of the circle whose length will be 2cm (or 2 radius length long), this bigger angle has a measure of 2 radians.3. If you'll make the second central angle of the circle bigger in order to intercept an arc whose length will be 3cm (or 3 radius length long), this angle has a measure of 3 radians.These three cases illustrate the following relationship:measure of an angle in radians = length of the intercepted arc/length of the radiusSo, if you label the angle with x, the length of the intercepted arc with s, and the length of the radius with r, you find the general formula: x = s/rIf you look at the picture of this circle you will see that in a half of the circle you can find a little more than 3 radians, and you need to go a little more far to make a full half of the circle. So, what is the number of radians that you need to go in order to make a full half of the circle?Let's look at this.Let's make a circle. If AB is a diameter of circle O with radius of length r, the points A and B separate the circle into two semicircles. Let's label the angle AOB with x, and the semicircle length with s.You will see that the angle AOB is a central angle of the circle, so the measure of this angle in radians is equal to the length of the intercepted arc AB, which is s, divided by the length of the radius which is r. (See the general formula above)You know that the circumference of the circle is:C = 2(pi)r, since s = C/2 we write:s = [2(pi)r]/2s = (pi)r, substitute this at the general formulax = s/rx = [(pi)r]/rx = piSince x is in radians we can say that in a half of the circle there is pi radians.But we know at the same time that in a half of circle there is 180 degrees, so we can say that the following relationship is true.180 deg = pi radNow you can see why is convenient to work with radians in a circle, because there is exactly pi radians in a half of the circle, and from this relationship we can see how radians are relating to degree measures.Now for a full circle how many degrees are?There are 360 degrees. How many radians it that will be?180 deg = pi rad360 deg = 2pi radSo there are 2pi rad in a full of a circle.Now you start to see why the circumference of a circle is 2(pi)r. Because we know that there are 2pi radius length arcs on a circle.How to convert degrees to radians?We know that:180 deg = pi rad Let's write this a little bit differently180(1 deg) = pi rad1 deg = (pi/180) rad So, what about 5 degrees? Just multiply both sides by 55(1 deg) = 5(pi/180) rad What about x deg?So, let' go t the general formula:x(1 deg) = x(pi/180) radso you have the general formula to use it when you need to convert degrees to radiansHow to convert radians to degrees? pi rad = 180 deg pi(1 rad) = 180 deg 1 rad = 180/pi deg What about 5 radians? Just multiply both sides by 5 5(1 rad) = 5(180/pi) deg What about x radians? So let's go to the general formula: x(1 rad) = x(180/pi) degSo you have the general formula to use it when you need to convert radians to degrees.
