Video: Finding the Central Angle Subtended by an Arc given the Arc’s Measure as a Fraction of the Circle’s Circumference

An arc measures 53/120 of the circumference of a circle. What angle does the arc subtend at the center?

02:04

Video Transcript

An arc measures fifty-three one hundred and twentieth of the circumference of a circle. What angle does the arc subtend at the centre?

So, let’s start this question by drawing our circle. We’re given an arc. And we’re asked for the angle that the arc subtends at the centre. So, if we draw the sector that this arc forms part of, then the angle that we’re looking for is the angle of the sector, which we can call 𝜃. We’re given a fractional measure for the measurement of the arc, fifty-three one hundred and twentieth. This means that if the circumference is 120, the arc would measure 53 or a proportional equivalent.

For example, 106 over 240 would be the case if the circumference was 240 then the arc would measure 106. We can write that the proportion of the angle 𝜃 over the sum of the angles at the centre is equivalent to the arc length over the circumference. So, to find the sum of the angles at the centre of a circle, we can recall the fact that the sum of the measures of the central angles in a circle is 360 degrees. We can now write our equation as 𝜃 over 360 equals 53 over 120.

So, to find 𝜃, we would take the cross product or cross multiply. We can start by working out 𝜃 times 120, which we can write as 120𝜃. This will be equal to 360 times 53. Simplifying the right-hand side will give us 120𝜃 equals 19080. To find 𝜃, we divide both sides of our equation by 120. And since 19080 divided by 120 is 159, then we have our final answer for the angle of the arc 159 degrees.

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