How far does the tip of a 5-inch minute hand travel in 20 minutes? The answer is 10.47 inches, and you can find it in one line. Arc length and sector area both come from a single idea: a slice of a circle is a fraction of the whole. Learn that idea once, and you can solve forward, backward, and on real objects like wipers and running tracks.
- Find the fraction first: the central angle divided by 360 degrees.
- Arc length equals that fraction times the circumference, 2 x pi x r.
- Sector area equals that same fraction times the full area, pi x r squared.
- In radians the formulas shrink to s = r x theta and A = r squared x theta / 2.
- To work backward, rearrange: theta = s / r, and r = square root of 2A / theta.
How Do You Calculate Arc Length and Sector Area?
Divide the central angle by 360, then multiply that fraction by the circumference for arc length and by the full area for sector area. The same fraction drives both answers.
Take a circle with radius 8 cm and a 45 degree angle. The fraction is 45 / 360, which is exactly 1/8. The full circumference is 2 x pi x 8, or 50.27 cm, so the arc is 6.28 cm.
The full area is pi x 64, or 201.06 square cm. One eighth of that gives a sector area of 25.13 square cm. You never need a separate rule for each piece.
Our guide on how to find the circumference of a circle covers the full-circle step in detail. Here, you only scale it down.
A quick cross-check also exists. Sector area equals half the radius times the arc length, so 0.5 x 8 x 6.28 gives 25.13 again. Matching answers confirm the work.
Why Do Radians Make s = r x Theta So Simple?
A radian is defined as the angle whose arc equals the radius, so arc length is just the radius times the number of radians. No conversion factor is needed.
Look at the degree formula again: s = 2 x pi x r x (angle / 360). The piece 2 x pi / 360 converts degrees into radians. Once the angle is already in radians, that piece disappears.
The sector area formula shrinks the same way. The fraction theta / (2 x pi) times pi x r squared leaves r squared x theta / 2. Both formulas become one clean multiplication.
A full circle holds 2 x pi radians, or about 6.28. That is why a circumference is 2 x pi x r: you lay about 6.28 radius lengths around the edge.
Radian mode on a calculator, and when to switch it, is explained in our degrees vs radians calculator guide. Always check the mode before you trust a sine or arc result.
How Do You Work Backward to Find the Angle or Radius?
Rearrange the same formulas: theta = s / r gives the angle, and r = square root of (2A / theta) gives the radius. Convert radians to degrees at the end.
Angle From Arc and Radius
An arc of 15 cm sits on a circle of radius 6 cm. Divide 15 by 6 to get 2.5 radians. Multiply by 180 / pi, and the angle is 143.24 degrees.
Radius From Sector Area and Angle
A sector has an area of 50 square inches and a 72 degree angle. The fraction is 72 / 360, or 0.2, so the full circle holds 250 square inches. Divide by pi and take the square root to get a radius of 8.92 inches.
Angle From the Fraction
A sector covers 45 square units of a circle with a total area of 150. The fraction is 45 / 150, or 0.3. Multiply 0.3 by 360 degrees, and the central angle is 108 degrees.
The pattern stays the same each time. Find the fraction, then use it to recover the missing piece.
Why Is an Arc Always Longer Than Its Chord?
The chord is the straight line between the arc’s endpoints, and a straight line is the shortest path between two points. The curved arc must therefore be longer.
The gap depends on the angle. With radius 10, a 60 degree arc measures 10.47 while its chord is exactly 10.00. At 180 degrees, the arc is 31.42 and the chord is the diameter, 20, so the arc runs 57 percent longer.
Small angles hide a useful insight. At 10 degrees, the arc is 1.7453 and the chord is 1.7431, a gap of only 0.13 percent. For small slices, the chord is a near-perfect stand-in for the arc.
Segment Area: Sector Minus Triangle
A segment is the region between the arc and its chord. Take the sector area and subtract the triangle formed by the two radii and the chord.
For radius 10 and 60 degrees, the sector is 52.36 and the triangle is 0.5 x 100 x sin 60, or 43.30. The segment area is 9.06 square units.
How Do Arc Formulas Solve Everyday Problems?
Match the object to a radius and an angle, then apply the fraction. Garden edging and clock hands are two quick, common cases.
Curved Garden Bed Edging
A corner bed forms a 120 degree sector with a 6 foot radius. The fraction is 1/3, so the curved edge is 12.57 feet. The two straight sides add 12 feet, so buy about 24.57 feet of edging.
The same fraction sizes the soil. One third of pi x 36 gives 37.70 square feet of bed to cover with mulch.
A Clock’s Minute Hand
A minute hand sweeps 360 degrees in 60 minutes, so 20 minutes is 120 degrees, or 2.094 radians. With a 5 inch hand, the tip travels 5 x 2.094, or 10.47 inches.
The tip of a longer hand moves farther in the same time. Double the hand to 10 inches, and the tip covers 20.94 inches.
To check any of these answers in seconds, the arc length and sector area calculator returns the arc, sector, and chord from a radius and an angle.
How Do You Handle Ring-Shaped Sweeps Like Wipers and Tracks?
Treat the shape as an annular sector: find the outer sector, then subtract the inner sector. For lengths, compare the arcs at each radius.
Windshield Wiper Sweep
A wiper arm reaches 24 inches from its pivot, and the blade starts 6 inches out. It sweeps 110 degrees. The outer sector is 552.92 square inches, and the inner one is 34.56.
The cleaned area is the difference, 518.36 square inches. The blade tip travels 46.08 inches per sweep, while the inner end covers only 11.52.
Running Track Bend
A standard track bend is a half circle with an inner radius of about 36.5 meters. The half circle measures pi x 36.5, or 114.67 meters.
Each lane is 1.22 meters wide, so each lane outward adds pi x 1.22, or 3.83 meters, per bend. That growth is why outer runners start ahead of inner ones.
The Arc Length Calculator gives you arc length, sector area, and chord length in degrees or radians in one step.
FAQs About Arc Length and Sector Area
What Is the Difference Between Arc Length and Sector Area?
Arc length is the distance along the curved edge, measured in length units. Sector area is the space inside the wedge, measured in square units. Both use the same angle fraction of the circle.
Do I Need Radians to Find Arc Length?
No. With degrees, use s = 2 x pi x r x angle / 360. With radians, use s = r x theta. Both give the same answer when the angle is converted correctly.
How Do I Find the Central Angle From Arc Length?
Divide the arc length by the radius to get the angle in radians. Multiply by 180 / pi for degrees. An arc of 15 cm on a radius of 6 cm gives 2.5 radians, or 143.24 degrees.
How Do I Find the Radius From Sector Area?
Divide the sector area by the angle fraction to get the full circle area. Then divide by pi and take the square root. A 50 square inch sector at 72 degrees gives a radius of 8.92 inches.
Is the Chord Ever Longer Than the Arc?
No. The chord is a straight line, and a straight line is the shortest path between two points. The two lengths almost match at small angles, differing by 0.13 percent at 10 degrees.
How Do I Find the Area of a Circular Segment?
Subtract the triangle from the sector. The segment area equals r squared / 2 x (theta – sin theta), with theta in radians. For radius 10 at 60 degrees, the segment is 9.06 square units.
How Do I Find the Area of a Ring-Shaped Sector?
Find the sector area at the outer radius, then subtract the sector area at the inner radius. A 110 degree sweep from 6 to 24 inches covers 518.36 square inches.
Sources
References Used in This Article
This article is general geometry education. Worked answers are rounded to two decimal places unless noted. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 26, 2026.
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Shakeel Muzaffar is the Founder and Editor-in-Chief of MultiCalculators.com, bringing over 15 years of experience in digital publishing, product strategy, and online tool development. He leads the platform's editorial vision, ensuring every calculator meets strict standards for accuracy, usability, and real-world value. Shakeel personally oversees content quality, formula verification workflows, and the platform's commitment to publishing tools that are genuinely useful for students, professionals, and everyday users worldwide.
