Worksheets · Precalculus

Arc length and sector area worksheet for Precalculus

Precalculus works these forwards and backwards: given the arc length, find the radius or the central angle, and for a segment, subtract a triangle from a sector, so exact answers mix π with radicals. The angles here are in degrees, so the fraction of the circle is the angle over 360.

Also taught in Geometry and Algebra 2, each at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Arc Length and Sector Area

Date Period

Find each measure. Leave answers in terms of π.

  1. Find the arc length, angle in degrees

    A circle has radius 12. Find the length of a 30∘ arc.
  2. Find the length of the arc cut off by a 180∘ central angle in a circle with diameter 8.
  3. Find the length of the arc cut off by a 90∘ central angle in a circle with radius 6.
  4. Find the arc length for a 30∘ arc of a circle with diameter 12.
  5. Find the arc length for a 180∘ arc of a circle with radius 5.

  1. Find the sector area, angle in degrees

    Find the area of the sector with a 120∘ central angle in a circle with diameter 12.
  2. A circle has radius 8. Find the area of a 45∘ sector.
  3. Find the sector area for a 60∘ sector of a circle with diameter 12.
  4. Find the sector area for a 120∘ sector of a circle with radius 9.
  5. A circle has diameter 20. Find the area of a 90∘ sector.

  1. Find the central angle or the radius

    An arc of a circle of radius 8 is 32π3 long. Find its central angle.
  2. An arc with a central angle of 270∘ is 12π long. Find the radius.
  3. An arc of a circle of radius 9 is 9π long. Find its central angle.
  4. An arc with a central angle of 72∘ is 18π5 long. Find the radius.
  5. An arc of a circle of radius 6 is π long. Find its central angle.

  1. Find the area of a segment

    A chord splits a circle with radius 4 into two arcs, one of them 90∘. Find the exact area of the region bounded by the chord and that arc.
  2. Find the exact area of the segment formed by a 60∘ arc and its chord in a circle with radius 2.
  3. A chord of a circle with radius 14 subtends a central angle of 120∘. Find the exact area of the segment between the chord and its arc.
  4. In a circle with radius 4, a chord joins the ends of a 30∘ arc. Find the exact area of the region between the chord and that arc.
  5. The arc cut off by a chord in a circle with radius 18 measures 60∘. What is the exact area of the segment between them?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Arc Length and Sector AreaAnswer keyVersion 3

Date Period

Find each measure. Leave answers in terms of π.

  1. Find the arc length, angle in degrees

    A circle has radius 12. Find the length of a 30∘ arc.2π
  2. Find the length of the arc cut off by a 180∘ central angle in a circle with diameter 8.4π
  3. Find the length of the arc cut off by a 90∘ central angle in a circle with radius 6.3π
  4. Find the arc length for a 30∘ arc of a circle with diameter 12.π
  5. Find the arc length for a 180∘ arc of a circle with radius 5.5π

  1. Find the sector area, angle in degrees

    Find the area of the sector with a 120∘ central angle in a circle with diameter 12.12π
  2. A circle has radius 8. Find the area of a 45∘ sector.8π
  3. Find the sector area for a 60∘ sector of a circle with diameter 12.6π
  4. Find the sector area for a 120∘ sector of a circle with radius 9.27π
  5. A circle has diameter 20. Find the area of a 90∘ sector.25π

  1. Find the central angle or the radius

    An arc of a circle of radius 8 is 32π3 long. Find its central angle.240∘
  2. An arc with a central angle of 270∘ is 12π long. Find the radius.8
  3. An arc of a circle of radius 9 is 9π long. Find its central angle.180∘
  4. An arc with a central angle of 72∘ is 18π5 long. Find the radius.9
  5. An arc of a circle of radius 6 is π long. Find its central angle.30∘

  1. Find the area of a segment

    A chord splits a circle with radius 4 into two arcs, one of them 90∘. Find the exact area of the region bounded by the chord and that arc.4π−8
  2. Find the exact area of the segment formed by a 60∘ arc and its chord in a circle with radius 2.23π−3
  3. A chord of a circle with radius 14 subtends a central angle of 120∘. Find the exact area of the segment between the chord and its arc.1963π−493
  4. In a circle with radius 4, a chord joins the ends of a 30∘ arc. Find the exact area of the region between the chord and that arc.43π−4
  5. The arc cut off by a chord in a circle with radius 18 measures 60∘. What is the exact area of the segment between them?54π−813

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

An arc of a circle of radius 8 is 6π long. Find its central angle

  1. The whole circumference is 2π × 8 = 16π.
  2. The arc is 6π out of 16π, which is 3/8 of the circle.
  3. 3/8 of 360° is 135°.

The answer is 135∘.

Where students go wrong

Forgetting the fraction of the circle when working backwards. A 135° arc that is 6π long is 3/8 of the circumference, so the whole circle is 16π and r = 8, not 3.

Also called sector area, length of an arc or area of a sector.

What you can put on this worksheet

Arc length, angle in degrees
Find the arc length for a 90∘ arc of a circle with radius 6. → 3π
Sector area, angle in degrees
Find the sector area for a 90∘ sector of a circle with radius 6. → 9π
Arc length, angle in radians
Find the arc length: r=5, θ=3π2. → 15π2
Sector area, angle in radians
Find the sector area: r=5, θ=3π2. → 75π4
The radius or the angle, working backwards
An arc of a circle of radius 8 is 6π long. Find its central angle. → 135∘
Area of a segment
A circle has diameter 24. A chord cuts off a 90∘ arc. Find the area of the segment between the chord and that arc, exactly. → 36π−72
Read off a drawn sector, round to a tenth
Find the arc length of the shaded sector. Round to the nearest tenth. — a circle with a shaded sector, labeled 12 cm, 240° → 50.3 cm

Questions about these worksheets

Yes. Every arc length and sector area sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the arc length and sector area questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 7: arc length, angle in degrees, sector area, angle in degrees, arc length, angle in radians, sector area, angle in radians, the radius or the angle, working backwards, area of a segment and read off a drawn sector, round to a tenth. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the fraction of the circle for the arc length but forgetting it on the area, or the other way round. Both are the same fraction of the whole; only the whole is different.