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Circles: Radians, arcs and sectors
Measuring angles in radians, circumference & arc lengths and sector areas
Measuring angles in radians, circumference & arc lengths and sector areas
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The ratio of a circle's perimeter to its diameter is constant in all circles. This constant is called π (pi). Since the diameter is twice the radius, the circumference of a circle is
The area of a circle is
where r is the radius of the circle.
A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:
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The area of a circle is πr2, and there are 360 degrees of rotation in a circle. Therefore, a sector with central angle θ is 360°θ of a full circle, and has area
One radian is the interior angle of an arc which has a length equivalent to the radius r of the circle. Since the circumference of a circle is given by 2πr, then, there are 2π total radians in a circle (the equivalent of 360°).
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Since the perimeter of a full circle is 2πr, the angle θ corresponding to a full circle (360°) is
So
Some key angles in radians and degrees:
An arc is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.
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Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, the arc length is calculated as
A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:
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The area of a circle is πr2, and there are 2π radians in a circle. Therefore, a sector with central angle θ is 2πθ of a full circle, and has area
so