Geometry
How To Calculate The Area Of A Sector
A sector is the wedge-shaped piece of a circle bounded by two radii and the arc between them. Its area is a simple fraction of the whole circle, with the fraction set by the central angle.
Quick Answer
sector area = (angle / 360) x pi x r^2, or 0.5 x r^2 x theta in radians
- r
- The radius of the circle
- angle
- The central angle, in degrees
- theta
- The same angle in radians
- sector area
- The wedge-shaped area
Convert the angle to radians, then take half the radius squared times that angle. A radius of 5 with a 90 degree angle gives 19.634954084936208 of area, an arc of 7.853981633974483, and a quarter of the full circle area of 78.53981633974483.
What Is The Area Of A Sector?
A sector is the shape you get when you cut a pie. Two straight edges run from the centre to the crust, and the curved edge between them is an arc. The angle at the centre between the two straight edges is the central angle, and it decides how much of the circle the wedge covers.
The area follows directly from the fraction. A full circle is 360 degrees, so a wedge with a central angle of 90 degrees covers 90 divided by 360, which is one quarter, of the whole circle. Multiply that fraction by the circle area of pi times the radius squared and you have the sector area.
Radians make the formula cleaner. A full turn is 2 pi radians, so a wedge of theta radians covers theta divided by 2 pi of the circle. That works out to an area of 0.5 times the radius squared times theta, with no 360 to carry around.
The two formulas always agree. For a radius of 5 and a 90 degree angle, the fraction method gives 0.25 times 78.53981633974483, which is 19.634954084936208. The radian method gives 0.5 times 25 times 1.5707963267948966, which is the same 19.634954084936208.
The arc length is the curved edge itself, and it is the radius times the angle in radians. For a radius of 5 and a 90 degree angle that is 5 times 1.5707963267948966, or 7.853981633974483, which is a quarter of the full circumference of 31.41592653589793.
A semicircle is the special case of a 180 degree angle, and a quarter circle is 90 degrees. Both show up often enough to be worth recognising on sight: a semicircle of radius 5 has an area of 39.269908169872416, and a quarter circle has 19.634954084936208.
The sector and the triangle formed by the same two radii are related. That triangle has an area of 0.5 times the radius squared times the sine of the angle, which is always a little less than the sector area, and the difference is the little curved sliver between the straight chord and the arc.
Sectors appear in real design work wherever a round object is cut or measured in pieces. Lawn sprinklers cover sectors, radar and sonar sweeps are sectors, and pie charts are sectors stacked side by side.
Units matter as much as the angle. The angle is dimensionless, so the area carries the square of whatever unit the radius is in. A radius of 5 metres with a 90 degree angle gives 19.634954084936208 square metres, and the arc is measured in plain metres.
Checking your work is easy. Add up the areas of all the sectors you have cut and the total must equal the full circle area. If a quarter and a half are cut, the remaining quarter is 19.634954084936208 for a radius of 5, and nothing has been lost or double counted.
Formula
sector area = (angle / 360) x pi x r^2
Formula page →The wedge's share of the circle, taken straight from the angle as a fraction of a full turn.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| angle | Central angle | degrees | The angle at the centre, in degrees. |
| r | Radius | any length unit | Distance from the centre to the arc. |
| A | Sector area | square length units | The area of the wedge. |
| 360 | Full turn | degrees | The degrees in one complete circle. |
sector area = 0.5 x r^2 x theta, where theta = angle x pi / 180
Formula page →The same result with the angle in radians, which removes the division by 360.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| theta | Angle in radians | radians | angle x pi / 180. |
| r | Radius | any length unit | Distance from the centre to the arc. |
| A | Sector area | square length units | The area of the wedge. |
| 0.5 | Half | unitless | The constant that makes the units come out right. |
arc length = r x theta
Formula page →The curved edge of the sector, which is the radius times the angle in radians.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| s | Arc length | any length unit | The curved edge of the sector. |
| r | Radius | any length unit | Distance from the centre to the arc. |
| theta | Angle in radians | radians | The central angle in radians. |
How To Calculate The Area Of A Sector
- 1
Read the central angle
The angle must be the one at the centre between the two radii, not an angle somewhere on the circle. A 90 degree angle cuts a quarter of the circle.
- 2
Convert to radians if you prefer the short formula
Multiply the degrees by pi and divide by 180. Ninety degrees becomes 1.5707963267948966 radians.
- 3
Compute the full circle area
Pi times the radius squared. For a radius of 5 that is 78.53981633974483, which is the total you are taking a fraction of.
- 4
Take the fraction
Divide the angle by 360 and multiply by the circle area, or use 0.5 times the radius squared times the radians. Either route gives 19.634954084936208.
- 5
Check against the full circle
The sector must be smaller than the whole circle, and the fractions of all your sectors must add to one. A quarter circle of radius 5 is exactly 19.634954084936208.
Examples
Example 1: A radius of 5 with a 90 degree angle
- Radius
- 5
- Central angle
- 90
| Step | Calculation | Result |
|---|---|---|
| Angle in radians | 90 x pi / 180 | 1.5707963267948966 |
| Full circle area | pi x 5 x 5 | 78.53981633974483 |
| Fraction of the circle | 90 / 360 | 0.25 |
| Sector area | 0.25 x 78.53981633974483 | 19.634954084936208 |
| Arc length | 5 x 1.5707963267948966 | 7.853981633974483 |
Result: A radius of 5 with a 90 degree central angle gives a sector area of 19.634954084936208 and an arc length of 7.853981633974483, which is exactly a quarter of the full circle area of 78.53981633974483.
Example 2: A radius of 10 with a 45 degree angle
- Radius
- 10
- Central angle
- 45
| Step | Calculation | Result |
|---|---|---|
| Angle in radians | 45 x pi / 180 | 0.7853981633974483 |
| Full circle area | pi x 10 x 10 | 314.1592653589793 |
| Fraction of the circle | 45 / 360 | 0.125 |
| Sector area | 0.125 x 314.1592653589793 | 39.269908169872416 |
| Arc length | 10 x 0.7853981633974483 | 7.853981633974483 |
Result: A radius of 10 with a 45 degree central angle gives a sector area of 39.269908169872416 and an arc of 7.853981633974483, which is one eighth of the full 314.1592653589793 circle.
Example 3: A radius of 3 with a 270 degree angle
- Radius
- 3
- Central angle
- 270
| Step | Calculation | Result |
|---|---|---|
| Angle in radians | 270 x pi / 180 | 4.71238898038469 |
| Full circle area | pi x 3 x 3 | 28.274333882308138 |
| Fraction of the circle | 270 / 360 | 0.75 |
| Sector area | 0.75 x 28.274333882308138 | 21.205750411731103 |
| Arc length | 3 x 4.71238898038469 | 14.137166941154069 |
Result: A radius of 3 with a 270 degree central angle covers three quarters of the circle, giving a sector area of 21.205750411731103 and an arc of 14.137166941154069.
Calculator
Sector area
19.635
- Arc length
- 7.854
- Share of the full circle
- 25.00%
- Full circle area
- 78.5398
- Central angle in radians
- 1.5708
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Area Of A Sector calculator page.
Common Mistakes
Using the width of the wedge instead of the central angle
The formula needs the angle at the centre between the two radii. An angle measured somewhere else on the circle gives a completely different fraction.
Mixing degrees and radians
The 0.5 times radius squared formula needs radians. Feeding it 90 degrees gives an area 57 times too large, because 90 radians is more than fourteen full turns.
Forgetting to square the radius
The radius appears squared in every circle area formula. Using 0.5 times 5 times 1.5707963267948966 gives 3.9 instead of 19.634954084936208.
Dividing by 2 pi instead of 360 when using degrees
The fraction method uses degrees over 360. Dividing 90 by 6.283185307179586 gives a fraction of 14.3, which is nonsense and produces an area far larger than the circle.
Reporting the arc as the area
Arc length is a straight distance while sector area is square units. For a radius of 5 at 90 degrees those are 7.853981633974483 and 19.634954084936208 respectively.
Assuming a sector can be larger than the circle
The angle caps the fraction at 1, so a 90 degree wedge can never exceed 78.53981633974483 for a radius of 5. If it does, the angle has been misread.
FAQ
What is the area of a sector?
It is the central angle divided by 360, multiplied by pi times the radius squared. For a radius of 5 at 90 degrees that is 19.634954084936208.
How do I find the arc length of a sector?
Multiply the radius by the angle in radians. For a radius of 5 at 90 degrees that is 5 times 1.5707963267948966, or 7.853981633974483.
What is the area of a quarter circle?
A quarter circle is a 90 degree sector, so it is one quarter of pi times the radius squared. For a radius of 5 that is 19.634954084936208.
What is the area of a semicircle?
A semicircle is a 180 degree sector, half the circle area. For a radius of 5 that is 39.269908169872416.
What is the relationship between sector area and arc length?
Sector area is half the radius times the arc length, because area is 0.5 times radius squared times theta and arc is radius times theta. Both scale with the angle.
Can I use the diameter instead of the radius?
Only if you halve it first. Every circle formula is written in terms of the radius, so a diameter of 10 becomes a radius of 5 and the sector area is 19.634954084936208.
References
- [1]Wikipedia, Circular sector — https://en.wikipedia.org/wiki/Circular_sector
- [2]Wikipedia, Radian — https://en.wikipedia.org/wiki/Radian
- [3]Wikipedia, Arc (geometry) — https://en.wikipedia.org/wiki/Arc_(geometry)