| CIRCLES: ARCS and SECTORS |
Central Angles - Circumference Arc
Circular Clock Face
In 2 minutes, the time on this clock will be 15 minutes to 4 o'clock. Another way to say this time is a quarter to 4. We call 15 minutes "a quarter of an hour" because the minute hand rotates ¼ of the way around the circular face in that time. We know that 360° is a full circle, and we also know that a full circle is an hour or 60 minutes. So 90° or ¼ of 360°, corresponds to 15 minutes which is ¼ of 60 minutes.
The length of the circumference arc, traced by the minute hand as it moves through a 90° angle, is exactly ¼ the length of the clockface circumference. If we know the length of the minute hand (radius), we can find the length of the arc it traces. It will measure ¼ the length of the circumference.
| The length of a sector arc is proportional to the sector angle. |
We find the length of an arc created by a central angle of a circle with this proportion
Remember that circumference is 
so we have to know the diameter or radius of the circle in order to find its circumference. If the minute hand on the clock is 1.5 cm. long, we find the distance its tip travels in 15 minutes with this proportion:
since 90° is ¼ of 360°, the arc is ¼ the circumference or 
Example:
Find the length of the arc traced by a 3 cm. minute hand in 24 minutes.
Solution:
The circumference of the face is: 
24 minutes is 
of an hour or 360°,
so the arc length = 
The ratio of arc length to circumference equals the ratio of the central angle to 360°.
Sometimes, we're given the length of the arc and we have to find the central angle.
Example:
A circle has radius = 11 inches. How big is the central angle that cuts off an arc of 23.4 inches?
Solution:
If we know the ratio of Arc to Circumference, we know the ratio of central angle to 360°. We first find the circumference:
Now we set up the proportion. We'll use x to represent the measure of the central angle.
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Central Angles - Sector Area
In the same way that the size of a sector arc is proportional to the central angle that subtends it, the ratio of sector area to circle area equals the ratio of central angle to 360°.
Example
We find the area of a sector created by a central angle of a circle with this proportion
Example:
A circle has radius = 11 inches. Find the area of a sector created by a central angle of 20°.
Solution:
We find the circle's area:
Now we set up the proportion. We'll use x to represent the sector area.
In some questions, we know the area, so we need to find the central angle. We use the proportion statement but the variable now is in the angle position instead of the sector area spot.
Example
The area of a sector is one-fifteenth the area of the whole circle. How big is the central angle?
Solution
Since the sector area is 1/15 the circle's area, the angle must be 1/15 of 360°.
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Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.
Practice Exercise
Make a diagram if you're stuck. It really helps!
1)
a) How long is the arc formed by a 37° central angle in a circle with radius = 3.6 inches.
b) What size central angle subtends an arc of 3.77 cm in a circle with circumference = 18.85 cm?
c) Angle AOB = 65°. AO is a radius = 7.3 cm. What is the area of sector AOB?
2) An arc of 27.3 cm is cut off by a 45.6 o central angle in a circle.
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Solutions
1)
a) Perimeter = 22.62 inches so
The arc is 2.32 inches long.
b) What size central angle subtends an arc of 3.77 cm in a circle with circumference = 18.85 cm?
The angle is 72 °.
c) Angle AOB = 65°. AO is a radius = 7.3 cm. What is the area of sector AOB?
The area of a circle with r = 7.3 is 167.42 cm².
The area of the sector is 30.23 cm².
2)
a) We use a proportion since C = arc cut off by 360°
So, 
b) What is the radius of the circle?
Since 
c) What is the area of the circle?
d) What is the area of the sector?
So,
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