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Geometry - Circles, Arcs, and Sectors

6 important questions on Geometry - Circles, Arcs, and Sectors

Repeated question (from previous section), there are 2 ways to talk about the 'size' of an arc. What are they?

1) Arc measure (the number of degrees in an arc)
2) Arc length

Of the 2 ways to talk about arc 'size', what would be the motivation to choose on over another aka when do you use arc measure to talk about arc size and when do you use arc length ?

Same circle vs. Different circles
When comparing arcs within the same circle, arc measure is fine.
Comparing arcs between different circles, arc measure is not enough and we use arc length .

How do we find the length of an arc?

Look at the proportion of the part over the whole (part/whole)

For degrees its:
arc measure (in degrees) / total degrees in a circle, 360

That proportion must also be true for length:
length of arc / total length of circle (which is the circumference).

So once we know the proportion, we can apply that to the part we want to find out.

Can we divide the whole area of a circle into fractions?

Yes, such a 'slice' of a circle is called a circular sector.

What is a circular sector?

A circular sector is when you divide the are of a circle into fractions. One such fraction is called a circular sector.

How do we find the area of a circular sector?

We already know how to find the arc length in a circle, using a part-to-whole proportion.

From here we can do the same but instead of using the circumference we use the area as the denominator.:

Proportion
angle/360

must be equal to:
area of sector/ area of circle