Area of a Sector Calculator (from Radius and Central Angle)

Enter the radius and the central angle of the sector. The area (πr² × central angle ÷ 360) is calculated and a figure of the sector is drawn.

Units

Radius r

Central angle (degrees)

Central angle (radians)

Enter the central angle in only one field, degrees or radians. Enter numbers only, without units (enter the radius in cm and the area is in cm²; in m, it is in m²).

Result and figure

Enter the radius and the central angle (in degrees or radians, only one of them) in the fields on the left and press "Calculate". The result and a figure of the sector will appear here.

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What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
- Area of a sector (central angle in degrees)
- Area of a sector (central angle in radians)
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the radius and the central angle, and you get the area of the sector (\(\pi r^2 \times\) central angle \(\div\, 360\)) right away
The central angle can be entered in degrees (°) or in radians (fill in only one of the two fields)
When it divides evenly, the exact answer in terms of \(\pi\) is shown too (for a radius of 30 and a central angle of 90°, \(225\pi\))
A figure of the sector is drawn with the result, so you can see at a glance which part of the whole circle you found the area of
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

The calculation uses \(\pi = 3.14159\ldots\) (the "3.14" used in school is a rounded value of it). Enter numbers only, without units. Enter the radius in inches and the area is in in²; in feet, it is in ft².

What is this calculation used for?

Comparing the size of a slice of pizza or cake (sharing food)

Cut a 14-inch pizza (radius 7 in) into 8 slices, and each slice is a sector with a central angle of \(360^\circ \div 8 = 45^\circ\). Its area is \(\pi \times 7^2 \times \dfrac{45}{360} = \dfrac{49\pi}{8} \approx 19.24\,\mathrm{in^2}\).
Questions like "Which is bigger, a slice of a medium cut into 8 or a slice of a large cut into 12?" can be settled with numbers by finding the areas with this formula.

Estimating the area a sprinkler covers (gardening and farming)

A rotating sprinkler waters a sector whose radius is how far the water reaches and whose central angle is how far it turns. A sprinkler that reaches 30 ft and turns 180° covers \(\pi \times 30^2 \times \dfrac{180}{360} = 450\pi \approx 1{,}413.7\,\mathrm{ft^2}\).
Compare this with the size of your lawn or garden, and you can plan how many sprinklers you need and where to put them before you buy (wind and water pressure change the actual coverage, so treat it as a rough guide).

Working out the area a windshield wiper cleans (car design and maintenance)

A wiper swings in a sector around the base of its arm. If the tip reaches 24 in and it swings through 90°, the sector it sweeps is \(\pi \times 24^2 \times \dfrac{90}{360} = 144\pi \approx 452.4\,\mathrm{in^2}\) (about 3.14 ft²).
The part actually wiped is this sector minus the small inner sector the blade does not reach (a shape like a piece of a ring), so you use this formula twice: "large sector − small sector".

Finding the area of a fan-shaped flower bed or plaza (landscaping and gardening)

Fan-shaped flower beds and lawns are common at street corners and in front of buildings. A bed with a radius of 10 ft and a central angle of 120° has an area of \(\pi \times 10^2 \times \dfrac{120}{360} = \dfrac{100\pi}{3} \approx 104.7\,\mathrm{ft^2}\).
Sod, soil and mulch are sold by how much area they cover (a bag covers so many ft², and so on), so once you know the area, you can estimate how much to buy.

Checking the area a security camera or sensor covers (home security)

Specs for security cameras and motion sensors say things like "90° field of view, 25 ft detection range". Treating the area covered at ground level as a sector gives \(\pi \times 25^2 \times \dfrac{90}{360} = 156.25\pi \approx 490.9\,\mathrm{ft^2}\).
Lay this over a plan of a parking lot or a store, and you can check for blind spots and how many units you need before installing (obstacles and the tilt of the camera change the area it actually sees).

Formulas and figures

Area of a sector (central angle in degrees)

Figure

Standard notation (the usual math form)

\(S\) \(=\) \(\pi\) \(\times\) \(r\) \(2\) \(\times\) \(\theta\) \(\div\) \(360\)

In words (symbols replaced with words)

⑥ \(S\): area of the sector \(=\) ③ \(\pi\): pi \(\times\) ① \(r\): radius ② squared (the length times itself) \(\times\) ④ \(\theta\): central angle (degrees) \(\div\) ⑤ \(360^\circ\): angle of the whole circle

The formula in words

① Take the \(r\): radius

② and square it (the length times itself)

③ then multiply by \(\pi\): pi (about 3.14) to get the area of the whole circle, \(\pi r^2\). Next,

④ multiply by the \(\theta\): central angle (degrees)

⑤ and divide by the \(360^\circ\): angle of the whole circle (this multiplies by the fraction of the whole circle that the central angle covers)

⑥ and you get the \(S\): area of the sector

Quick example

The area of a sector with a radius of 30 in and a central angle of 90° is

\(S\): area of the sector \(=\) \(\pi\): pi \(\times\) radius (30 in) squared \(\times\) central angle (90°) \(\div\) whole circle (360°)

\(\pi \times 30^2 \times \dfrac{90}{360} = 900\pi \times \dfrac{1}{4} = 225\pi\)

\(225\pi = 225 \times 3.14159\ldots \approx 706.86\,\mathrm{in^2}\)

Key idea

A sector is a slice cut from a circle at the central angle, like a slice of pizza. So its area is just the area of the whole circle, \(\pi r^2\), times the fraction of the whole circle (360°) that the central angle covers, \(\left(\dfrac{\theta}{360}\right)\). A central angle of 90° is \(\dfrac{90}{360} = \dfrac{1}{4}\) of the circle, 180° is half, and 360° is exactly the whole circle. When the central angle divides 360° evenly (90°, 60°, 45° and so on), simplifying \(\dfrac{\theta}{360}\) first makes the calculation easier.

Area of a sector (central angle in radians)

Figure

Standard notation (the usual math form)

\(S\) \(=\) \(\pi\) \(\times\) \(r\) \(2\) \(\times\) \(\theta\) \(\div\) \(2\pi\)

In words (symbols replaced with words)

⑥ \(S\): area of the sector \(=\) ③ \(\pi\): pi \(\times\) ① \(r\): radius ② squared (the length times itself) \(\times\) ④ \(\theta\): central angle (radians) \(\div\) ⑤ \(2\pi\): angle of the whole circle

The formula in words

① Take the \(r\): radius

② and square it (the length times itself)

③ then multiply by \(\pi\): pi (about 3.14) to get the area of the whole circle, \(\pi r^2\). Next,

④ multiply by the \(\theta\): central angle (radians)

⑤ and divide by the \(2\pi\): angle of the whole circle (radians) (this multiplies by the fraction of the whole circle that the central angle covers)

⑥ and you get the \(S\): area of the sector

Quick example

The area of a sector with a radius of 10 in and a central angle of 1.5708 radians (about 90°) is

\(S\): area of the sector \(=\) \(\pi\): pi \(\times\) radius (10 in) squared \(\times\) central angle (1.5708 radians) \(\div\) whole circle (\(2\pi\) radians)

\(\pi \times 10^2 \times \dfrac{1.5708}{2\pi} = 10^2 \times \dfrac{1.5708}{2}\)

\(100 \times 1.5708 \div 2 = 78.54\,\mathrm{in^2}\)

Key idea

Radians are a way of measuring angles taught in high school (radian measure): \(180^\circ = \pi\) radians and \(360^\circ = 2\pi\) radians. The idea is exactly the same as with degrees: multiply the area of the whole circle, \(\pi r^2\), by the fraction of the whole circle (\(2\pi\) radians) that the central angle covers. In this formula the \(\pi\) on top and the \(\pi\) on the bottom cancel out, so it simplifies to the short form found in textbooks, \(S = \dfrac{1}{2} r^2 \theta\) (radius squared × central angle ÷ 2).

The area of a sector is "the area of the whole circle \(\pi r^2\) × the fraction of the circle the central angle covers". For a central angle in degrees, divide by 360; in radians, divide by \(2\pi\) (which simplifies to \(S = \dfrac{1}{2} r^2 \theta\)). Enter the radius in inches and the answer is in in².

Symbols and terms

Symbols

\(S\)	ess	A symbol often used for area. On this page it is the area of the sector. (Many US textbooks write area as \(A\).)
\(r\)	ar	The radius, from the first letter of "radius". It is the length from the center of the circle to the arc.
\(r^2\)	r squared	The number \(r\) multiplied by itself (\(r \times r\)). The small 2 at the upper right is an exponent that tells you to multiply the number by itself.
\(\theta\)	theta	A Greek letter often used for the size of an angle. On this page it is the central angle of the sector.
\(\pi\)	pi	Pi - the number of times the circumference of a circle is longer than its diameter. It is \(3.14159265\ldots\), a decimal that goes on forever without repeating (an irrational number). In school it is often rounded to 3.14.
\(^\circ\)	degree	A unit of angle. One full turn divided into 360 equal parts is 1°. It is the most familiar way to write an angle, the one you measure with a protractor (degree measure).
\(\mathrm{rad}\)	radian	A unit of angle taught in high school (radian measure). The central angle of an arc as long as the radius is 1 radian, and \(180^\circ = \pi\) radians (1 radian ≈ 57.3°).

Terms

sector	A shape cut from a circle by two radii, like a slice of pizza or an open folding fan. It is bounded by two radii and an arc.
central angle	The angle at the point of the sector, that is, at the center of the circle. It shows how wide the sector opens; the larger it is, the larger the area and the arc.
arc	Part of the circumference of a circle. It is the curved outer edge of the sector.
pi	The value of circumference ÷ diameter. It is the same for a circle of any size (\(3.14159\ldots\)). The symbol is \(\pi\).
degree measure	Measuring angles with one full turn as 360°. It is the familiar degree (°) measure used since elementary school.
radian measure	Measuring angles by arc length (the unit is the radian). One full turn is \(2\pi\) radians. It is used from Algebra 2 and precalculus on.
fraction of the whole circle	A number that shows how much of the whole something takes up. On this page, "central angle ÷ angle of the whole circle" shows what fraction of the whole circle the sector is.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.

Fractions of a whole and multiplying fractions (Grades 5–6)	Being able to write "how much of the whole" as a fraction, such as \(\dfrac{90}{360} = \dfrac{1}{4}\) Being able to simplify fractions and multiply a number by a fraction
Area of a circle (Grade 7)	Knowing that the area of a circle is "radius × radius × pi (3.14)" Being able to read and write units of area such as in² and ft²
Expressions with variables and pi (Grade 7)	Being able to leave pi as the letter \(\pi\) and write an answer such as \(225\pi\)
Arcs and sectors (high school Geometry)	Knowing the terms sector, central angle and arc Being able to picture a sector as "a piece cut from a circle in proportion to the central angle"
Radian measure (Algebra 2 and precalculus; only if you use the radians field)	Knowing what a radian is and that \(180^\circ = \pi\) radians (one full turn = \(2\pi\) radians)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the area of a sector (central angle in degrees)

Radius	30
Central angle (degrees)	90
Area of the sector	=PI()B1^2B2/360

Table to find the area of a sector (central angle in radians)

Radius	10
Central angle (radians)	1.5708
Area of the sector	=B1^2*B2/2

After pasting, column A has the item names and column B the numbers. The upper rows are your inputs, and the formula in the last row calculates from them automatically.
"B1" and "B2" in a formula stand for "the number in that cell". "*" is multiplication, "/" is division and "^" is a power (how many times to multiply). "PI()" is a function that returns pi (3.14159…).
In the first table, B3 shows π × 30² × 90 ÷ 360 ≈ 706.86 (in² if you entered inches). The second table (the radian version, radius² × central angle ÷ 2) shows 78.54 in B3. Just replace the inputs with the numbers for your own sector.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the area of a sector (central angle in degrees)

Radius	30
Central angle (degrees)	90
Area of the sector	=PI()B1^2B2/360

Table to find the area of a sector (central angle in radians)

Radius	10
Central angle (radians)	1.5708
Area of the sector	=B1^2*B2/2

The same formulas as in Excel (including the PI() function) work as is. Copy the whole table, paste it into cell A1, and replace the inputs with the numbers for your own sector.

How to calculate it in Python

import math

radius = 30    # radius (inches in this example)
degrees = 90   # central angle (degrees)

area = math.pi * radius ** 2 * degrees / 360   # area of the sector (input unit squared; in2 in this example)

print(f"Area of the sector: {area} in2")

# With the central angle in radians, it is "radius squared x central angle / 2"
radians = 1.5708
area_by_radians = radius ** 2 * radians / 2
print(f"With a central angle of {radians} radians: {area_by_radians} in2")

Runs with the standard library only. "math.pi" is pi, "**" is a power (squaring here), "*" is multiplication and "/" is division. Change the radius and the central angle at the top and run it.

How to write it in LaTeX and other math languages (copy and paste)

Area of a sector (central angle in degrees)

S = πr² × θ ÷ 360

S = \pi r^{2} \times \dfrac{\theta}{360}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>S</mi>
    <mo>=</mo>
    <mi>&#x03C0;</mi>
    <msup><mi>r</mi><mn>2</mn></msup>
    <mo>&#xD7;</mo>
    <mfrac><mi>&#x03B8;</mi><mn>360</mn></mfrac>
  </mrow>
</math>

S = pi r^2 xx theta/360

Pi r^2 theta/360

S := Pi*r^2*theta/360;

S = pi*r^2*theta/360;

S = πr^2 × θ/360

Area of a sector (central angle in radians)

S = πr² × θ ÷ (2π)

S = \pi r^{2} \times \dfrac{\theta}{2\pi}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>S</mi>
    <mo>=</mo>
    <mi>&#x03C0;</mi>
    <msup><mi>r</mi><mn>2</mn></msup>
    <mo>&#xD7;</mo>
    <mfrac><mi>&#x03B8;</mi><mrow><mn>2</mn><mi>&#x03C0;</mi></mrow></mfrac>
  </mrow>
</math>

S = pi r^2 xx theta/(2 pi)

Pi r^2 theta/(2 Pi)

S := Pi*r^2*theta/(2*Pi);

S = pi*r^2*theta/(2*pi);

S = πr^2 × θ/(2π)

How to have ChatGPT do the calculation

You are a math calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

A sector has a radius of 30 in and a central angle of 90°.
Find each of the following:
1. The area of this sector in in² (as a decimal)
2. The same area written exactly in terms of pi (π)

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula