Sector area is one of 6 circles formulas in the geometry section of this library, and it is used at high school level.
Why sector area works
A sector is a slice of the whole disk, and the share it takes is the same as the share its angle takes of a full turn. Multiplying the whole disk area by the angle divided by 2π cancels one π and one factor of 2, leaving half the radius squared times the angle.
What each symbol means
$r$ is radius and $\theta$ is the central angle in radians.
Sector area: when it holds
$\theta$ must be in radians; for degrees use $A=(\theta/360^\circ)\pi r^2$.
When it stops applying
It measures the pie slice bounded by two radii and the rim. The smaller piece cut off by a straight chord is not the same thing, and you get that one by taking away the triangle formed by the two radii.
Sector area: a worked example
For $r=3$ and $\theta=\pi/2$, $A=9\pi/4$.
The mistake to avoid
What people do: Using the rim-length rule when the question asked for the region.
Why it goes wrong: That answer measures a distance along the curved edge, so it arrives in plain length units where square units were needed.
Do this instead: Square the radius and halve it for a region; keep the plain product of radius and angle for the curved edge alone.
Sector area: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r$ is radius and $\theta$ is the central angle in radians.
- Check the conditions before substituting. $\theta$ must be in radians; for degrees use $A=(\theta/360^\circ)\pi r^2$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Circles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where sector area comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Equation of a Circle — the lesson behind this formula: read and build circle equations on the coordinate plane.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about sector area
What share of the disk does a right angle cut off?
One quarter. A right angle is a quarter of a full turn, so the slice covers a quarter of the whole disk.
How do I do this when the angle is in degrees?
Put the angle over 360 and multiply that fraction by the area of the whole disk, which avoids converting to radians at all.
How does this relate to the length of the curved edge?
The slice covers half the radius times that curved length, which is the circular version of half base times height.
Can the angle be larger than a straight angle?
Yes. A reflex angle up to a full turn describes the larger piece of the disk, and the formula handles it with no change.