An arc is a fraction of the circumference and a sector is the same fraction of the area, and the fraction is the angle over 360. That is the whole method in degrees. The radian versions are one option away: there the arc length is just r times theta, which is the reason radians exist.
Also taught in Algebra 2 and Precalculus, 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.
Find the arc length for a 60∘ arc of a circle with radius 9
60 over 360 is one sixth.
The circumference is 18 pi.
One sixth of 18 pi.
The answer is 3π.
Where students go wrong
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.
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=23π.→ 215π
Sector area, angle in radians
Find the sector area: r=5,θ=23π.→ 475π
The radius or the angle, working backwards
An arc of a circle of radius 8is 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.
Geometry is usually taken around 9th or 10th 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.