You are watching: How many radians in half a circle
Angles space measured in standard position from the optimistic horizontal axis walk counter-clockwise (for the positive direction). One radian is defined to be the angle so that the arc that the unit one subtended by the angle has actually length 1 (one radian is about 57.2958°). Since the circumference of the unit one is 2πr = 2π (the radius the the unit circle is r = 1) and the angle is in direct proportion to the arc length, one whole change measures 2π radians. The one is then divided up to find the radian measure up of various other nice angles.
Half a change is because of this equal come 2π/2 = π radians. One quarter of a revolution (a right angle) equates to 2π/4 = π/2 radians. Three quarters of a revolution measures 3 × π/2 = 3π/2 radians.
One eighth the a change is the angle that divides the first quadrant in half; the is half a right angle: (π/2)/2 = 2π/8 = π/4 radians. The angle that bisect the other quadrants space π/2 + π/4 = 3π/4 radians, π + π/4 = 5π/4 radians and 3π/2 + π/4 = 7π/4 radians.
The edge one third of the way through the first quadrant is one twelfth the a revolution, the is, (π/2)/3 = 2π/12 = π/6 radians. Dual π/6 is two-thirds that the method through the an initial quadrant, or one 6th of a revolution: 2 × π/6 = 2π/6 = π/3 radians. The angle one third and two thirds that the way through the second, 3rd and 4th quadrants room π/2 + π/6 = 3π/6 + π/6 = 2π/3 radians, π/2 + π/3 = 3π/6 + 2π/6 = 5π/6 radians, π + π/6 = 7π/6 radians, π + π/3 = 4π/3 radians, 3π/2 + π/6 = 9π/6 + π/6 = 5π/3 radians and 3π/2 + π/3 = 9π/6 + 2π/6 = 11π/6 radians.
Finally, let"s not forget the angle through measure 0 radians.
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The "nice" angle in boosting order: 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4, 11π/6, 2π.
Of course, friend can continue to subdivide one revolution to locate any type of angle friend wish. Angles with measure greater than 2π finish one or much more complete revolution(s) plus the remainder left when dividing the angle measure up by 2π. Angle with negative measure room taken in the clockwise direction.