How To Find Arc Length Of A Circle Without Radius

Arc Measure Definition

An arc is a segment of a circle around the circumference. An arc measure is an angle the arc makes at the middle of a circle, whereas the arc length is the span forth the arc. This angle measure tin be in radians or degrees, and we can easily catechumen between each with the formula $π r a d i a n south = 180 °$ .

You can also measure out the circumference, or distance effectually, a circumvolve. If you take less than the full length around a circle, bounded by two radii, you have an arc. That curved piece of the circumvolve and the interior space is called a sector, like a piece of pizza. When you cut up a circular pizza, the chaff gets divided into arcs.

Arc Measure out Definition
Arc of a Circle
Arc Mensurate vs. Arc Length
Degrees and Radians
Arc Measure out Formula
How To Find The Mensurate of an Arc
How To Discover Arc Length
Identifying Arc Bending Indicated

Arc of a Circle

If nosotros cutting beyond a delicious, fresh pizza, we take two halves, and each half is an arc measuring $180 °$ . If we brand three additional cuts in i side only (so we cut the half outset into two quarters and then each quarter into ii eighths), we have i side of the pizza with 1 large, $180 °$ arc and the other side of the pizza with iv, $45 °$ arcs like this:

$Measure Of An Arc$

The half of the pizza that is ane giant slice is a major arc since it measures $180 °$ (or more than). The other side of the pizza has four minor arcs since they each measure out less than $180 °$ .

Arc Measure vs. Arc Length

The arc is the fraction of the circle's circumference that lies between the two points on the circle. An arc has two measurements:

$Arc Measure vs Arc Length$

The arc'southward length is a distance along the circumference, measured in the same units every bit the radius, diameter or entire circumference of the circle; these units volition be linear measures, similar inches, cm, m, yards, and and so on
The arc'due south angle measurement, taken at the centre of the circle the arc is role of, is measured in degrees (or radians)

Practise non confuse either arc measurement (length or bending) with the directly-line distance of a chord connecting the ii points of the arc on the circle. The chord's length volition always be shorter than the arc's length.

Degrees and Radians

To be able to summate an arc measure out, you need to understand angle measurements in both degrees and radians. An angle is measured in either degrees or radians. A circle measures 360 degrees, or $2 π r a d i a n s$ , whereas one radian equals 180 degrees. So degrees and radians are related by the following equations:

$360 ° = two π r a d i a n southward$

$180 ° = π r a d i a n s$

The human relationship between radians and degrees allows us to catechumen to one another with uncomplicated formulas. To convert degrees to radians, we take the degree measure multiplied by pi divided by 180.

Let's convert 90 degrees into radians for instance:

$90 ° \times (\frac{π}{180 °})$

$\frac{90 π}{180}$

$\frac{π}{2} radians$

Now let'south convert $\frac{π}{iii} r a d i a north s$ to degrees:

$\frac{π}{3} \times \frac{180}{π}$

$\frac{180 π}{3 π}$

$\frac{180}{three} = 60 °$

Arc Measure Formula

Once you got the hang of radians, we can use the arc measure formula which requires the arc length, $s$ , and the radius of the circle, $r$ , to calculate.

$a r c m e a s u r east = \frac{a r c l e due north yard t h}{r a d i u south} = \frac{s}{r}$

How To Find The Measure of an Arc

Allow's attempt an example where our arc length is 3 cm, and our radius is 4 cm equally seen in our illustration:

$Arc Measure Formula$

Start with our formula, and plug in everything we know:

$a r c 1000 e a s u r e = \frac{s}{r}$

$a r c m e a s u r eastward = \frac{three}{4}$

Now we can catechumen $\frac{iii}{4} r a d i a n s$ into degrees past multiplying by 180 dividing by $π$ .

$(\frac{3}{4}) (\frac{180}{π})$

$42.9718 \approx 43 °$

How To Observe Arc Length

You demand to know the measurement of the fundamental angle that created the arc (the angle of the two radii) to calculate arc length. The arc length is the fractional corporeality of the circumference of the circumvolve. The circumference of any circle is found with $2 π r$ where $r = r a d i u s$ . If yous have the diameter, you tin can also utilise $π d$ where $d = d i a m eastward t e r$ .

The formula for finding arc length is:

$A r c l eastward north yard t h = (\frac{a r c a due north g fifty e}{360 °}) (two π r)$

Let'south try an case with this pizza:

$How To Measure Arc Length Of A Circle$

Our pie has a diameter of 16 inches, giving a radius of 8 inches. Nosotros know the slice is $sixty °$ . So the formula for this particular pizza slice is:

$= \frac{lx °}{360 °} \cdot 2 \cdot π \cdot 8$

$= \frac{ane}{6} \cdot 16 π$

$\approx 8.3775 "$

Identifying Arc Angle Indicated

An arc angle'south measurement is shown every bit $thou \overset{⌢}{A B}$ where $A$ and $B$ are the two points on the circle creating the arc. The $one thousand$ ways measurement, and the short curved line over the $\overset{⌢}{A B}$ indicates we are referring to the arc. The two points derived from the central angle (the angle of the two radii emerging from the center point).

One important distinction between arc length and arc bending is that, for ii circles of dissimilar diameters, same-angle sectors from each circumvolve volition non accept the same arc length. Arc length changes with the radius or diameter of the circle (or pizza).

Lesson Summary

At present that you have eaten your way through this lesson, you tin can place and ascertain an arc and distinguish between major arcs and minor arcs. You are also able to measure an arc in linear units and degrees and use the correct symbol, $1000 \overset{⌢}{A B}$ (where A and B are the two points on the circumvolve), to evidence arc length.