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 arcis 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° central angle in a circle.

- a) What is the circumference of the circle?

- b) What is the radius of the circle?

- c) What is the area of the disc?

- d) What is the area of the sector?

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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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( *Plane Geometry MathRoom Index* )

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