Arc measure Definition

An arc is a segment the a circle approximately the circumference. An arc measure is an edge the arc provides at the facility of a circle, vice versa, the arc length is the expectancy along the arc. This angle measure deserve to be in radians or degrees, and also we can conveniently convert between each with the formula π radians = 180°.

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You can also measure the circumference, or distance around, a circle. If girlfriend take much less than the full length approximately a circle, bounded by 2 radii, you have actually an arc. That curved piece of the circle and also the interior space is called a sector, favor a part of pizza. When you reduced up a circular pizza, the late gets divided into arcs.

Arc of a Circle

If us cut throughout a delicious, new pizza, we have two halves, and each half is an arc measure up 180°. If us make three added cuts in one side only (so we cut the half an initial into two quarters and then every quarter right into two eighths), we have one next of the pizza through one big, 180° arc and also the various other side that the pizza with four, 45° arcs prefer this:


The fifty percent of the pizza the is one gigantic slice is a major arc because it procedures 180° (or more). The other side of the pizza has 4 minor arcs since they every measure much less than 180°.

Arc measure up vs. Arc Length

The arc is the portion of the circle"s circumference the lies in between the two points ~ above the circle. One arc has actually two measurements:


The arc"s length is a distance along the circumference, measure in the very same units together the radius, diameter or whole circumference of the circle; these devices will be linear measures, favor inches, cm, m, yards, and also so on The arc"s angle measurement, taken in ~ the center of the circle the arc is part of, is measure up in degrees (or radians)

Do not confuse one of two people arc measure up (length or angle) with the straight-line distance of a chord connecting the two points the the arc on the circle. The chord"s size will always be much shorter than the arc"s length.

Degrees and also Radians

To be able to calculate one arc measure, you require to understand angle measurements in both degrees and also radians. An angle is measure in either levels or radians. A circle procedures 360 degrees, or 2π radians, whereas one radian amounts to 180 degrees. For this reason degrees and radians are associated by the adhering to equations:

360° = 2π radians

180° = π radians

The relationship between radians and degrees enables us to transform to one one more with an easy formulas. To transform degrees to radians, us take the level measure multiplied by pi divided by 180.

Let"s transform 90 degrees into radians because that example:

90° × π180°


π2 radians

Now let"s convert π3 radians come degrees:

π3 × 180π


1803 = 60°

Arc measure Formula

Once you gained the hang of radians, we can use the arc measure up formula which calls for the arc length, s, and also the radius of the circle, r, to calculate.

How To find The measure of one Arc

Let"s shot an example where ours arc size is 3 cm, and our radius is 4 centimeter as checked out in our illustration:


Start through our formula, and also plug in whatever we know:

arc measure = sr

arc measure = 34

Now we can convert 34 radians into degrees by multiply by 180 dividing by π.


42.9718 ≈ 43°

How To find Arc Length

You need to know the measure up of the central angle that created the arc (the edge of the 2 radii) to calculation arc length. The arc length is the fractional amount of the one of the circle. The one of any kind of circle is found with 2πr wherein r = radius. If you have the diameter, girlfriend can likewise use πd whereby d = diameter.

The formula because that finding arc size is:

Let"s shot an example with this pizza:


Our pie has a diameter of 16 inches, giving a radius the 8 inches. We understand the slice is 60°. Therefore the formula for this specific pizza part is:

      = 60°360° · 2·π·8

      = 16 · 16π

      ≈ 8.3775"

Identifying Arc edge Indicated

An arc angle"s measurement is displayed as mAB⌢ where A and B space the two points top top the circle developing the arc. The m method measurement, and also the brief curved line end the AB⌢ shows we are referring to the arc. The 2 points derived from the central angle (the edge of the two radii emerging from the facility point).

One important distinction between arc length and arc angle is that, for 2 circles of various diameters, same-angle sectors from each circle will certainly not have the same arc length. Arc length alters with the radius or diameter that the circle (or pizza).

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Lesson Summary

Now that you have consumed your means through this lesson, you can identify and also define one arc and also distinguish between significant arcs and also minor arcs. You are also able to measure up an arc in linear units and degrees and also use the correct symbol, mAB⌢ (where A and B room the two points ~ above the circle), to present arc length.

Next Lesson:

Tangent come a Circle

What you learned:

After working your means through this lesson and also video, you will discover to:

Identify and define one arc differentiate between significant arcs and minor arcs measure up an arc in straight units and also degrees use correct symbols to present arc size