A central angle of 1 radian is the angle that subtends an arc equal in length to the radius. If diameter = 5 m, then radius = 2.5 m. 2.5 m --> 1 radian 6 m is subtended by (6 / 2.5) = 2.4 radians.
Suppose the radius of the circle is r units and the sector subtends an agle of x radians at the centre of the circle. ThenArea = 0.5*r2*x square units.If x is measured in degrees, this becomesArea = pi*r2*x/360 square units.
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It is 10/18 = 0.55... radians.
Arch length, or linear distance of an arch, is equal to: (Angle of Arch in Radians) x (Radius of Arch) So for a complete circle (Arch angle of 360 Degrees) with a diameter of 3 feet (or radius of 1.5 feet) the linear circumference would be: (2 x PI) x (radius) = (2 x PI) x (1.5 ft), where PI = 3.14 radians and represents 1/2 of a complete circle. This is also the equation for a circle's circumference.
45 degrees are pi/4 radians. You can verify this with a unit circle.
If a triangle is drawn in a circle with a diameter as the base of the triangle, then the angle opposite that diameter is a right angle. This is an extension of the theorem that the angle which an arc of a circle subtends at the centre of a circle is twice the angle which the arc subtends at the circumference. In the case of a diameter, then the angle subtended at the centre is 180° and thus the angle at the circumference is 90°.
The length of an arc of a circle of radius r, which subtends an angle of x radians at the centre is r*x.
101.6 degrees = 1.7733 radians. So arc = radius*angle (in radians) = 219/2*1.7733 = 194.2 ft.
The answer is easier if angles are measured in radians. At the end, you'll get a conversion from radians to degrees.You are given a circle with diameter d, and a chord that subtends an arc of length x. Let c be the length of the chord and write r = d/2 for the radius of the circle.The arc subtends an angle of y = x/r radians at the centre of the circle.(If you draw a chord and the perpendicular from the centre of the circle to the midpoint of the chord, the following trigonometry is easier to understand).sin(y) = (c/2)/r where c is the length of the chord.so sin(y) = c/2rand so c = 2r*sin(y) = d*sin(y) = d*sin(x/r) = d*sin(x/2d)Everything on the right hand side is known and so c is easily calculated.Radians-degreesA whole circle is 360 degrees of 2*pi radians so 1 radian = 180/pi degrees. Writing 180/pi each time is tedious and also, higher maths is all in radians anyway, so I tend to avoid working with degrees. If you are using this on a calculator, make sure the angles are being measured in radians. The default units for angles in Excel is radians.
Suppose the radius of the circle is r units and the sector subtends an agle of x radians at the centre of the circle. ThenArea = 0.5*r2*x square units.If x is measured in degrees, this becomesArea = pi*r2*x/360 square units.
The answer depends on the information that you have. If the arc subtends an angle of x radians in a circle with radius r cm, then the arc length is r*x cm.
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Arc length is equal to radius times the angle the arc subtends (makes) at the centre of the circle, but the angle needs to be in radians. Set your calculator to radians instead of degrees, or, to change degrees to radians, divide by 180 and times pi. The formula comes from the fact that the length of the arc is proportional to the circumference of the circle in the same ratio as the angle at the centre is to the complete revolution at the centre, so length of arc: circumference of circle = angle size : 360o arc/(2*pi*r) = angle in degrees/360 or angle in radians/(2*pi) so arc length is angle in degrees divided by 360, times the circumference of the circle. Answer will be in the same measurement unit as the radius.
The length of an arc which subtends an angle of x radians at the centre of a circle with radius r cm is simply rx cm. If you measure angles in degrees then it will be r*x*(pi/180) cm
The relation between the arc of length and the central angle is that the arc of length divided by one of the sides is the central angle in radians. If the arc is a full circle, then the central angle is 2pi radians or 360 degrees.
2pi/9 radians or 40 degrees
The radian system describes angles in terms of the diameter of a unit circle, i.e. where the radius is 1. If two lines intersect at the radius of a unit circle, the angle in radians between those two lines is the length of the arc along the diameter of the circle delimited by those two lines. The diameter of a unit circle is 2 pi. In the degree system, the angle of one quarter of the circle is 90, while the radians of that same angle is pi / 2. One radian is approximately 57.3 degrees.