How to find the angle for a percentage of a circle - ACT Math

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Question

The sector pictured above is of the circle. What is the angle measure for the sector?

Answer

A question like this is very easy. You merely need to find out what is of the total degrees in a circle. This is:

$0.45 * 360 =162^{\circ}$ . That is it!

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Question

The area of sector is $80\pi:cm^2$ . This figure is not drawn to scale.

What is the measure of the angle of the sector?

Answer

You know that the area of a circle is computed by the equation:

$A=\pi r^2$

For our data, this is:

$A=\pi 40^2$ or $A=1600\pi:cm^2$

Now, the sector is a percentage of the circle. For the areas, this can be represented as the fraction:

$\frac{80\pi:cm^2}{1600\pi:cm^2}=\frac{1}{20}$

The total degree measure of a circle is, of course, degrees. This means that the sector contains:

$360^{\circ} * \frac{1}{20}=18^{\circ}$ .

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Question

The arc length of sector above is $4\pi:in$ . This figure is not drawn to scale.

What is the angle measure of sector ?

Answer

You know that the circumference of a circle is computed by the equation:

$C=2\pi r$

For our data, this is:

$C=28\pi=16\pi:in$

Now, the sector is a percentage of the circle. For the lengths of the circumference and the arc length, this can be represented as the fraction:

$\frac{4\pi:in}{16\pi:in}=\frac{1}{4}$

The total degree measure of a circle is, of course, degrees. This means that the sector contains:

$360^{\circ} \frac{1}{4}=90^{\circ}$ .

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Question

Sector is of the total circle. This figure is not drawn to scale.

What is the angle of this sector?

Answer

Do not overthink this question! All you need to remember is that a given circle contains degrees. This means that the sector is merely a percentage of . For our question, this percentage is , which is the same as . So, to calculate, you merely need to multiply:

$0.34 * 360^{\circ} = 122.4^{\circ}$

This is the degree measure of the sector.

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Question

A bike wheel has evenly spaced spokes spreading from its center to its tire. What must the angle be for the spokes in order to guarantee this even spacing? Round to the nearest hundredth.

Answer

Remember that the total degree measure of a circle is $360$^{\circ}$$ . This means that if you have parts into which you have divided your circle, each spoke must be $\frac{360}{15}$ or $24$^{\circ}$$ apart.

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