Son güncelleme: 19 Ağustos 2026
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Oluşturanlar
Dharmendra Singhİnceleyenler
Oluşturanlar
Dharmendra Singhİnceleyenler
Quick Answer
Sector area uses the degree-based formula A = (θ/360)πr², which treats the sector as the same fraction of the circle's area as the central angle is of 360 degrees. This calculator also returns arc length, chord length, and percent of the full circle to make the geometry easier to interpret.
The Sector Area Calculator multiplies the full circle area by the angle fraction θ over 360, then shows the matching arc length and chord length.
Önemli Noktalar
- Sector area is a fraction of the full circle area.
- The degree formula is A = (θ/360)πr².
- The same central-angle fraction also drives arc length.
- Chord length is a straight-line measure, not a curved one.
- Additional outputs help interpret the sector beyond area alone.
Oluşturanlar
Dharmendra Singhİnceleyenler
Oluşturanlar
Dharmendra Singhİnceleyenler
Formül
A = (θ / 360) × πr²
Burada:
- A=Sector area(square units)
- θ=Central angle(degrees)
- r=Radius(units)
Çözülmüş örnekler
Radius 6, angle 60°
A standard sector from a circle with radius 6.
- 1Use A = (θ/360)πr² with θ = 60 and r = 6.
- 2Compute the fraction of the circle: 60/360 = 1/6.
- 3Multiply by the full circle area 36π to get 6π ≈ 18.849556.
Radius 10, angle 90°
A quarter-circle sector.
- 1A 90° sector is one quarter of a full circle.
- 2The full area is 100π.
- 3One quarter of 100π is 25π ≈ 78.539816.
Radius 5, angle 120°
A sector covering one third of the circle.
- 1Compute the fraction 120/360 = 1/3.
- 2The full circle area is 25π.
- 3One third of 25π is 25π/3 ≈ 26.179939.
Giriş
The Sector Area Calculator finds the area of a slice of a circle when you know the radius and the central angle in degrees. Because a sector is simply a fraction of a full circle, the main idea is straightforward: compute what fraction of 360° the angle represents, then apply that same fraction to the area of the full circle. This page also returns arc length, chord length, and the percent of the full circle so the result is easier to interpret geometrically. That makes the calculator useful for geometry homework, design sketches, wheel and fan layouts, circular plots, and any situation where part of a circle matters more than the whole.
What a sector is
A sector is the region enclosed by two radii and the arc between them. If you picture a pizza slice, a fan blade sweep, or a pie-chart slice with curved outer edge, you are already picturing a sector. The geometry becomes easier when you see a sector as part of a whole circle rather than a completely separate shape.
Two radii form the straight sides.
The arc forms the curved side.
The central angle determines how much of the circle is included.
The radius controls the overall scale.
Why the formula works
The formula A = (θ/360)πr² works because the sector covers the same fraction of the full circle's area as the central angle covers of the full 360° turn. If the angle is 90°, the sector covers one quarter of the circle. If the angle is 120°, it covers one third. This proportional viewpoint is often easier to remember than the formula alone.
Find the angle fraction first.
Multiply the fraction by πr².
A larger radius increases area quadratically.
A larger angle increases area linearly.
Why arc length and chord length are included
Sector problems often lead to more than one useful output. Arc length tells you the length of the curved boundary, which matters for path length or material cutting. Chord length tells you the straight-line distance across the sector opening, which matters in construction and drawing. Seeing those outputs next to the area helps users distinguish surface measure from boundary measure.
Area measures region size.
Arc length measures curved boundary length.
Chord length measures the straight opening.
Percent of circle shows how big the slice is relative to the whole.
Worked example: radius 6 and angle 60°
Start with the fraction of the full circle. A 60° central angle represents 60/360 = 1/6 of the circle. The full circle area with radius 6 is π × 6² = 36π. One sixth of 36π is 6π, which is approximately 18.849556 square units. The same fraction applies to arc length: one sixth of the full circumference 12π gives 2π. The chord length is 6 because a 60° sector forms an equilateral-triangle relationship across the radii.
- Angle fraction:
1/6.
- Full circle area:
36π.
- Sector area:
6π.
- Arc length:
2π.
Common mistakes
The most common mistake is using the full circle area πr² and forgetting to multiply by the angle fraction. Another is mixing degrees and radians without noticing which formula version is being used. Users also sometimes confuse chord length with arc length even though one is straight and the other is curved. This calculator keeps both outputs visible so those quantities do not blur together.
Do not skip the θ/360 fraction.
Do not use a degree formula with a radian input.
Do not confuse curved arc length with straight chord length.
Do not forget that radius, not diameter, belongs in the formula.
When this calculation is useful
Sector-area work appears in school geometry, but it also shows up in practical layouts. Designers use it for circular panels and fan sweeps. Surveying and mapping can use sector reasoning for circular regions. Manufacturing can use it when cutting pie-shaped pieces from round stock. Even a pie chart or wheel slice becomes easier to reason about when you can move between angle, percent of circle, arc length, and area.
Geometry and trigonometry classes.
Circular design and drafting.
Material-cutting estimates.
Charts and visual data partitions.
Manual method versus calculator method
For clean benchmark angles like 60°, 90°, and 120°, solving by hand is excellent practice because the fractions are simple. The calculator becomes more useful when decimals are involved or when you want arc length and chord length at the same time. A good workflow is to estimate the slice fraction first, then calculate the exact outputs, and finally compare the area percentage with your visual intuition.
Estimate the fraction of the circle first.
Compute the area using the same fraction.
Use the additional outputs to check interpretation.
Keep the angle in degrees throughout if you use the degree formula.
Hızlı Referans Kartı
Sector area quick reference
Hızlı referans • Daire dilimi alanı hesaplayıcısı
A = (θ/360) × πr²Geçerli aralık: Use a positive radius and a central angle greater than 0° and at most 360°.
Yaygın Değerler
⚠ Dikkat
- •Use radius, not diameter, in the formula.
- •Keep the angle in degrees when using the θ/360 version.
- •Do not confuse arc length with chord length.
- •Angles above 360° do not describe one simple sector.
Pro İpuçları
- →Convert the angle to a fraction of 360° before doing any area arithmetic.
- →Estimate the percent of the circle first so the final area feels plausible.
- →Use benchmark angles like 60°, 90°, and 120° to check the calculator mentally.
- →Read arc length and chord length together if the problem mixes curved and straight boundaries.
SSS
What is the formula for sector area in degrees?
The degree-based formula is A = (θ/360)πr².
Why does the angle have to be divided by 360?
Because 360° represents the full circle, so θ/360 gives the fraction of the whole circle used by the sector.
Can the central angle be more than 360°?
Not for a single simple sector in this calculator. A full circle is already 360°, so larger values would describe multiple turns rather than one sector.
What is the difference between arc length and chord length?
Arc length follows the curved boundary of the sector, while chord length is the straight-line distance between the arc endpoints.
Why does the radius matter so much?
Area depends on r², so doubling the radius makes the corresponding sector area four times as large when the angle stays the same.
Can I solve sector area by hand?
Yes. Find the fraction θ/360 first, compute the full circle area πr², and then multiply the two.
When should I use the related arc-length calculator?
Use it when the curved boundary distance is the main quantity you need and area is only secondary or not needed at all.