finding the angle using the angle's rays and circumference of the circle

Question

A circle is centered at the vertex of an angle, and the angle's rays subtend an arc that is 87 cm long. 1/360^th of the circumference of the circle is 0.58 cm long. What is the measure of this angle in degrees?

A circle with a circumference of 504 cm is centered at the vertex of an angle, and the angle's rays subtend an arc that is 334.6 cm long. What is the measure of this angle in degrees?

Answer 1

Both of these questions can be solved by reasoning and proportions. With geometry, it often helps to draw a picture and you might need to search out some definitions for words you don't know like subtend.

For the first one, when it says 1/360th of the circumference is 0.58 cm, it is telling you that 1 degree corresponds to 0.58 cm because there are 360 degrees in a circle. So the question is, how many 0.58 cm are there in 87 cm? Which is a division questions so, 87 cm / 0.58 cm = 150 which we would interpret as 150 degrees

For the second one, the ratio of the arc to the circumference will be the same as the ratio of the angle to the to the 360 degrees in the entire circle. This translates to the equation 334.6 cm / 504 cm = x / 360. Solving for x, you get 239 degrees.

In general, I identify and setup proportions when I notice something like "this part is to this whole as that other part is to that other whole" which then translates to the equation: this part / this whole = other part / other whole.