SOLUTION: A circle has a chord of 25mm with an angle of pi/6 subtended at the centre. Find, to 1 dp, the length of the arc cut by the chord. 'm just not sure how to find the radius.

Click here to see ALL problems on Circles

Question 1138221: A circle has a chord of 25mm with an angle of pi/6 subtended at the centre. Find, to 1 dp, the length of the arc cut by the chord.
'm just not sure how to find the radius.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
see the attached diagram.
explanation is underneath.

the subtending central angle is angle BAC which is equal to pi/6 radians.

the chord is BC which is 25 millimeters long.

if you draw a line from the center of the circle intersecting with the chord at 90 degrees, this will split the chord in half.

see the bottom part of the diagram to see what that looks like.

that will make BD = 12.5 and DC = 12.5 millimeters in length.

you will form an isosceles triangle with AB and AC as the legs of it.

AB and AC are also radii of the circle.

the vertex of this isosceles triangle is split in half forming right triangles ADB and ADC.

angles BAD and CAD are each equal to half of angle BAC which makes them equal to pi/12 radians each.

sin(BAD) is equal to sin(pi/12) which is equal to opposite / hypotenuse of triangle ADB which make it equal to 12.5 / R.

solve for R to get R = 12.5 / sin(pi/12) which makes it equal to 48.29629131 millimeters in length.

the circumference of the circle is equal to 2 * pi * R = 303.454548 millimeters.

the arc cut by a central angle of pi/6 radians has a length of (pi/6) / (2pi) * the circumference of the circle which is equal to (pi/6) / (2pi) * 303.454548 = 25.287879 millimeters in length.

that's the length of arc BC. which is the length of the arc cut by the chord.

round to one decimal place and the length of arc BC = 25.3 millimeters.