Area of a Sector: Formula, Examples, and Calculation

You’ve probably been there — staring at a circle with a slice cut out, wondering how to find the area of that wedge without guessing. Whether you’re preparing for GCSE maths or just curious, the area of a sector formula is straightforward once you see it as a fraction of the whole circle.

Formula for area of a sector (degrees): θ/360 × πr² ·
Formula for area of a sector (radians): ½r²θ ·
Full circle area: πr² ·
Common sector angle examples: 90° (quarter circle), 60° (sixth circle), 45° (eighth circle) ·
Sector area for radius 5 cm, angle 60°: 13.09 cm² (approx)

Five key facts lay the foundation for every sector calculation you’ll encounter.

The pattern: these formulas all scale the full circle’s area by a fraction determined by the central angle.

What is the formula for the area of a sector?

What does each variable represent?

• θ (theta) = central angle of the sector, in degrees or radians
• r = radius of the circle
• πr² = area of the full circle

According to the GCSE maths revision guide from Third Space Learning, the standard degree formula is Area = (θ/360) × πr², where the fraction (θ/360) tells you how much of the circle you’re taking (Third Space Learning GCSE revision guide). For angles in radians, the maths education platform Cuemath gives Area = ½r²θ (Cuemath education platform).

How is the formula derived from the area of a circle?

• A full circle has 360° (or 2π radians).
• The sector’s area is proportional to its central angle relative to the full circle.
• So: sector area = (θ/360) × circle area.

Save My Exams AQA GCSE revision notes explain that sector and arc formulae are fractions of the circle formulas, with 360° representing the whole circle (Save My Exams revision notes).

What is the area of the sector?

What is a sector of a circle?

• A sector is a portion of a circle bounded by two radii and the arc between them.
• It looks like a pizza slice or a pie wedge.
• The area of a sector is proportional to its central angle.

Third Space Learning, a specialist maths tutoring provider, notes the area of a sector is a part of the full circle and uses the circle-area expression πr² in the calculation (Third Space Learning tutoring provider).

How to identify a sector

• Look for two straight lines (radii) meeting at the centre and a curved arc.
• The angle at the centre determines how large the sector is.
• If the angle is 90°, you have a quarter of the circle.

A GCSE video lesson emphasises that for a 90° sector, the area is one quarter of the area of a full circle (YouTube – GCSE Maths Tutor).

The implication: identifying the central angle is the critical first step. Without it, you cannot calculate the sector area.

What is the formula for the sector in GCSE maths?

How is the formula presented in GCSE exams?

• GCSE exam boards like AQA and Edexcel use the degree version: (θ/360) × πr².
• Angles are always given in degrees, never radians at Foundation tier.
• The formula may be given on the formula sheet, but you need to know how to use it.

Save My Exams’ AQA GCSE notes state: “To solve sector-area questions, divide the angle by 360, calculate the full circle’s area, and multiply by the fraction.” (Save My Exams revision notes).

What are common GCSE sector problems?

• Given radius and angle, find the sector area.
• Given sector area and angle, find the radius (reverse calculation).
• Find the area of a sector in a composite shape.

Third Space Learning’s GCSE revision guide includes a step-by-step method: find the radius, find the angle, substitute into the formula, and clearly state the answer (Third Space Learning GCSE revision guide). The same guide also provides practice exam questions and a free worksheet.

The pattern: GCSE questions rarely ask for the radian version. Stick with degrees and focus on fraction arithmetic.

What formula is 2 * pi * r?

What does 2πr represent?

• 2πr is the formula for the circumference of a circle (the distance around it).
• It uses the radius, but the result is a length, not an area.

Third Space Learning addresses this common confusion by separating the arc-length formula (θ/360) × πd from the sector area formula (Third Space Learning tutoring provider).

How does it differ from the area formula?

• Circumference is 2πr; area of a circle is πr².
• For a sector, you multiply the fraction by the full area (πr²), not by 2πr.
• If you use 2πr instead of πr², your answer will be a length, not an area — a classic exam mistake.

Third Space Learning also notes the perimeter of a sector is the arc length plus the two radii, which again involves circumference, not area (Third Space Learning tutoring provider).

The catch: many students confuse “perimeter” with “area” and apply the wrong formula.

How do I calculate the area of a sector?

Step-by-step calculation with degrees

1. Step 1: Write down the radius (r) and the central angle in degrees (θ).
2. Step 2: Substitute into the formula: Area = (θ/360) × πr².
3. Step 3: Calculate the fraction (θ/360) as a decimal or simplify.
4. Step 4: Multiply by πr² using a calculator (use π ≈ 3.14159 if needed).
5. Step 5: Round as required and state the units (e.g., cm²).

Third Space Learning’s four-step method emphasises this exact sequence (Third Space Learning GCSE revision guide). Example: radius = 5 cm, angle = 60°. Area = (60/360) × π × 5² = (1/6) × 78.54 ≈ 13.09 cm².

Step-by-step calculation with radians

1. Step 1: Ensure the angle is in radians. (If in degrees, convert by multiplying by π/180).
2. Step 2: Use formula: Area = ½ r² θ.
3. Step 3: Multiply r² by θ, then multiply by ½.
4. Step 4: State answer in square units.

Cuemath provides this radian-based form and highlights its simplicity compared to the degree version (Cuemath education platform).

How to use a calculator

• Enter the angle first, divide by 360 (or use the radian formula directly).
• Use the π button if available, else use 3.1415926535.
• Check that your calculator is in degree mode when using the degree formula.

A GCSE video lesson shows how to enter (θ/360) × πr² into a Casio calculator and get the correct decimal (YouTube – GCSE Maths Help).

Quick step-by-step recap

1. Identify the radius and the central angle.
2. Choose the correct formula based on angle units.
3. Substitute and compute.
4. Always include units in your answer.

Confirmed facts

• Area of sector = (θ/360)πr² for degrees (Third Space Learning GCSE revision guide)
• Area of sector = ½r²θ for radians (Cuemath education platform)
• Full circle corresponds to 360° or 2π radians (universal mathematical fact)

Expert perspectives on sector area

“Area of a sector = (θ/360) × πr², where θ is the central angle and r is the radius.”
– Third Space Learning (GCSE maths revision guide)

“To solve sector-area questions, divide the angle by 360, calculate the full circle’s area, and multiply by the fraction.”
– Save My Exams (AQA GCSE revision notes)

“For a 90° sector, the area is one quarter of the area of a full circle.”
– YouTube – GCSE Maths Tutor (video lesson)

Knowing the formula is only half the battle. The difference between a pass and a top grade often comes down to avoiding unit mistakes and checking your fraction arithmetic. For GCSE students sitting AQA or Edexcel papers, the standard degree formula is your safest bet — master it, and sector questions become simple proportion exercises.

For a quick check on your calculations, you can use a reliable interactive sector area calculator that works with both degrees and radians.

Frequently asked questions