Circle Calculator
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Result
Enter any one value. The calculator will find all other circle properties.
Enter a circumference to find radius and diameter, or enter a radius to find circumference.
Quick Examples
Circle Calculator: Area, Circumference, Arc Length & Circle Theorems
Your complete guide to solving circle problems, from basic area and circumference to arc length, sector area, chord length, and advanced circle theorems.
How to Use This Circle Calculator
Enter any one value (radius, diameter, area, or circumference). The calculator finds all other properties in one click. Great for quick homework checks.
Example: Enter radius = 5, get diameter = 10, area = 78.54, circumference = 31.42.
Enter the radius and central angle (degrees or radians). Get arc length, sector area, and chord length with step-by-step working.
Example: r = 10, θ = 90° → arc = 15.71, sector area = 78.54, chord = 14.14.
Convert circumference to radius and diameter, or go the other way. Useful when a problem gives you perimeter data and you need to find radius first.
Example: C = 31.42 → r = 5, d = 10.
Solve inscribed angle, central angle, tangent-chord, chord-chord, and coin rotation paradox problems. Select the theorem, enter the known angle or radius, and calculate.
Example: Inscribed angle = 40° → central angle = 80°.
Circle Calculator by Student Level
Topics: Area of a circle, circumference, diameter vs radius, and basic pi calculations. Problems typically give you the radius or diameter and ask you to calculate area or circumference.
How to use: Use the Circle tab. Enter the radius or diameter you are given and press Calculate. Read the step-by-step output to see how the formula A = πr² and C = 2πr are applied.
Tip: Always check whether the problem gives radius or diameter. Students often confuse the two and use the wrong value in the formula.
Topics: Arc length, sector area, chord length, inscribed angles, tangent-chord angles, chord-chord angles, and circle theorem proofs. These are common on GCSE Paper 2, IB SL, and SAT geometry sections.
How to use: Use the Arc & Sector tab for arc and sector problems. Use the Theorems tab for angle and chord problems. Work the problem on paper first, then verify your answer here and compare each step.
Tip: For GCSE and IB, you must often state the theorem name. Use the output labels (e.g., "Inscribed Angle Theorem") in your written answer for full marks.
Topics: Radian measure, arc integrals, polar coordinates, sector area in calculus, and rotational geometry including the coin rotation paradox. These topics appear in pre-calculus, Calculus II, and Discrete Math.
How to use: Switch the Arc & Sector tab to Radians for exact radian inputs. Use the Coin Rotation Paradox calculator to explore rotational geometry concepts. For polar area integrals, use the sector area formula as a building block.
Tip: Always convert to radians before applying L = rθ or A = ½r²θ. This calculator does the conversion for you when you select Degrees as the input unit.
Using This Tool for Circle Assignments
Homework Problems
Solve each problem by hand first, then enter the values here to verify your answer. If the numbers differ, compare your steps to the calculator output line by line. The most common homework mistakes involve using diameter instead of radius, forgetting to square the radius before multiplying by π, and using degrees instead of radians in arc formulas.
Case Studies and Projects
Applied math projects often use circles in real-world contexts, like calculating the area of a circular field, the length of a curved road, or the cross-section of a pipe. Enter the real measurements from your case study and use the step-by-step output to write up your solution in your report.
Quiz and Exam Preparation
Click Random to load a random example. Solve it on paper in under 60 seconds, then check. Repeat this 10 times before a test. Focus especially on the Arc & Sector tab since arc length and sector area are the most commonly tested circle topics at GCSE and SAT level.
Problem Sets
For multi-part problem sets, use the Circle tab to establish the radius first from whatever measurement is given, then move to the Arc & Sector or Theorems tab for the follow-up parts. Keeping the radius consistent across parts of a problem is key to avoiding carry-through errors.
What Is a Circle?
What is a circle in math?
A circle is a set of all points in a plane that are the same distance from a fixed point called the center. That fixed distance is called the radius. Every point on the circle is exactly one radius length away from the center. Circles appear in geometry, trigonometry, calculus, physics, and engineering.
What is the difference between radius and diameter?
The radius (r) is the distance from the center of the circle to any point on its edge. The diameter (d) is the distance across the circle passing through the center. The diameter is always exactly twice the radius: d = 2r. If a question gives you the diameter, divide it by 2 before plugging into area or arc formulas.
What is pi (π) and why does it appear in circle formulas?
Pi (π) is the ratio of a circle's circumference to its diameter. It equals approximately 3.14159 and is the same for every circle regardless of size. Because every circle formula relates a curved length or area back to the radius, π always appears as a constant multiplier. This calculator uses the precise value π = 3.14159265358979 for all computations.
Area of a Circle Calculator
What is the formula for the area of a circle?
The area of a circle is A = πr². You square the radius first, then multiply by π. For a circle with radius 6, the area is π × 36 = 113.10 square units. If you know the diameter d instead of radius, use A = π(d/2)² = πd²/4.
How do I find the radius if I know the area?
Rearrange A = πr² to get r = √(A/π). For an area of 50, the radius is √(50/3.14159) = √(15.915) ≈ 3.99 units. Use the Circle tab in this calculator, select "I know the Area," enter the value, and press Calculate to get the radius instantly.
Real-world example of finding circle area
A circular pizza has a diameter of 30 cm. What is its area? First, radius = 30/2 = 15 cm. Then area = π × 15² = π × 225 ≈ 706.86 cm². You can use this circle area calculator to verify the answer and see every multiplication step shown clearly.
Circumference Calculator
How do I calculate the circumference of a circle?
Use C = 2πr. Multiply the radius by 2, then multiply by π. For radius = 9, circumference = 2 × 3.14159 × 9 = 56.55 units. You can also write this as C = πd since d = 2r. Enter any radius in the Circle tab and the circumference appears instantly in the result panel.
How do I find the radius from the circumference?
Rearrange C = 2πr to get r = C / (2π). For a circumference of 44 units, r = 44 / (2 × 3.14159) = 44 / 6.2832 ≈ 7.00 units. Use the Circ ↔ Radius tab in this calculator. Select "Circumference to Radius," enter the circumference, and press Calculate.
What is the circumference-to-radius calculator useful for?
Real measurement problems often give you the perimeter of a round object instead of its radius. For example, a wheel has a circumference of 188.50 cm. What is its radius? r = 188.50 / (2π) = 30 cm. This type of reverse calculation is common in physics and engineering homework.
Arc Length Calculator
What is arc length?
Arc length is the distance along the curved edge of a circle between two points. It is part of the circumference. A full circle arc length equals the full circumference 2πr. A semicircle arc length is πr. For any partial arc, the formula is L = rθ, where θ is the central angle in radians.
How do I calculate arc length in degrees?
If the central angle is in degrees, use L = (θ/360) × 2πr. For a circle of radius 12 with a 45° arc, L = (45/360) × 2π × 12 = (1/8) × 75.40 = 9.42 units. In the Arc & Sector tab, simply select "Degrees" and the calculator converts for you automatically.
How do I solve arc length step by step?
Step 1: Identify the radius r and the central angle θ. Step 2: Convert θ to radians if it is in degrees by multiplying by π/180. Step 3: Apply L = rθ. For r = 8 and θ = 120°: θ in radians = 120 × (π/180) = 2.094 rad. L = 8 × 2.094 = 16.76 units.
Real-world example of arc length
A clock hand of length 10 cm sweeps through 90° from 12 to 3. How far does the tip travel? Arc length = (90/360) × 2π × 10 = (1/4) × 62.83 = 15.71 cm. Use the Arc & Sector tab with r = 10 and θ = 90° to verify this in one click.
Sector Area
What is a sector of a circle?
A sector is a pie-slice region bounded by two radii and the arc between them. It looks like a slice of pizza. The area of the sector depends on the radius and the central angle. A full circle is a 360° sector. A half-circle (semicircle) is a 180° sector. A quarter-circle is a 90° sector.
What is the sector area formula?
In radians: A = ½r²θ. In degrees: A = (θ/360) × πr². For a circle of radius 6 with a 60° sector, A = (60/360) × π × 36 = (1/6) × 113.10 = 18.85 square units. Enter r = 6 and θ = 60° in the Arc & Sector tab to see every step.
How is sector area different from arc length?
Arc length is a one-dimensional measurement in units (e.g., cm). Sector area is a two-dimensional measurement in square units (e.g., cm²). Arc length measures the curved edge of the slice. Sector area measures the filled region of the slice. Both depend on the same radius and central angle, but they use different formulas.
Chord Length
What is a chord in a circle?
A chord is a straight line connecting two points on the circumference of a circle. The longest chord passes through the center and is called the diameter. Any other chord is shorter than the diameter. Chord length depends on both the radius and the central angle between the two endpoints.
What is the chord length formula?
The chord length is c = 2r sin(θ/2), where r is the radius and θ is the central angle in radians. If the angle is in degrees, convert first. For a circle of radius 10 with a central angle of 60° (1.047 rad), chord = 2 × 10 × sin(30°) = 20 × 0.5 = 10 units. The Arc & Sector tab includes chord length in its output.
How do I solve for chord length in a GCSE problem?
GCSE problems often give you the radius and ask for the chord. Sometimes they give you the perpendicular distance from the center to the chord and ask you to use the Pythagorean theorem. For the direct formula approach, use c = 2r sin(θ/2). For the perpendicular bisector approach, use c = 2 × √(r² − d²) where d is the perpendicular distance.
Circle Theorems Calculator
What is the inscribed angle theorem?
The inscribed angle theorem states that an angle inscribed in a circle is half the central angle that subtends the same arc. If the central angle is 100°, any inscribed angle subtending the same arc is 50°. This is one of the most tested circle theorems in GCSE, IB, and SAT Math.
How do I find the central angle from an inscribed angle?
Central angle = 2 × inscribed angle. If the inscribed angle is 35°, the central angle is 70°. Use the Theorems tab, select "Inscribed Angle to Central Angle," enter 35, and press Calculate to verify. The output shows the multiplication step so you can see the formula applied.
What is the tangent-chord angle theorem?
A tangent-chord angle is formed between a tangent line and a chord at the point of tangency. The theorem states that this angle equals half the arc intercepted by the chord. If the intercepted arc is 80°, the tangent-chord angle is 40°. This is tested in GCSE and IB Higher papers.
What is the chord-chord angle theorem?
When two chords intersect inside a circle, the angle formed equals half the sum of the two intercepted arcs. If the two arcs are 80° and 60°, the angle = (80 + 60)/2 = 70°. Enter both arc values in the Theorems tab under "Chord-Chord Angle" to solve this instantly.
What is the coin rotation paradox?
The coin rotation paradox is a classic puzzle. If a small coin of radius r rolls around the outside of a larger coin of radius R without slipping, how many rotations does the small coin make? The intuitive answer is R/r, but the correct answer is R/r + 1. The extra rotation comes from the small coin traveling around the complete circle of the large coin. For two equal coins, it rotates twice, not once.
What are all the main circle theorems I need to know?
The core circle theorems are: (1) The angle at the center is twice the angle at the circumference subtending the same arc. (2) Angles in the same segment are equal. (3) The angle in a semicircle is always 90°. (4) Opposite angles in a cyclic quadrilateral add up to 180°. (5) The tangent to a circle is perpendicular to the radius at the point of contact. (6) Two tangents from an external point are equal in length. (7) The perpendicular from the center bisects a chord.
Real World Uses of Circle Calculations
Where is the area of a circle used in real life?
Area calculations appear whenever you deal with circular surfaces. Architects calculate the floor area of circular rooms. Engineers calculate cross-sectional areas of pipes. Farmers calculate irrigation coverage from sprinklers. Chefs calculate pizza sizes. Scientists calculate the area of a circular beam of light or sound.
Where is arc length used in real life?
Arc length describes the distance traveled along any curved path. A car turning along a circular road, a satellite orbiting Earth, a clock hand sweeping across time, and a roller coaster following a loop all involve arc length. In construction, curved walls and arched bridges require arc length calculations for material estimates.
Where are circle theorems used in real life?
Circle theorems are used in engineering and design. The inscribed angle theorem helps locate signal sources in radar and sonar. The tangent-chord property is used in gear design. Astronomers use arc relationships to measure angular distances between stars. Architects use cyclic polygon properties when designing domes and arches.
Frequently Asked Questions
- 1. How do you calculate the area of a circle?
- Use the formula A = πr². Square the radius first, then multiply by π (approximately 3.14159). If you know the diameter, divide by 2 to get the radius first. For a radius of 5, area = π × 25 = 78.54 square units. Use the Circle tab in this calculator, select "I know the Radius," enter 5, and press Calculate to see each step.
- 2. How do you calculate circumference from radius?
- Circumference = 2πr. Multiply the radius by 2π (approximately 6.2832). For radius = 7, circumference = 2 × 3.14159 × 7 = 43.98 units. Alternatively, if you know the diameter, use C = πd. Enter the radius in the Circle tab and the circumference is shown instantly alongside area and diameter.
- 3. How do I solve arc length step by step?
- Step 1: Write down the radius r and the central angle θ. Step 2: If the angle is in degrees, convert to radians: multiply by π/180. Step 3: Apply L = rθ. For r = 10 and θ = 90°: θ = 90 × (π/180) = 1.5708 rad. L = 10 × 1.5708 = 15.71 units. Use the Arc & Sector tab to verify with full step-by-step output.
- 4. What is the difference between arc length and sector area?
- Arc length (L = rθ) is the distance along the curved edge of the sector, measured in units like cm or m. Sector area (A = ½r²θ) is the area of the pie-slice region, measured in square units like cm². Both formulas use the same radius and central angle in radians. The Arc & Sector tab calculates both at the same time.
- 5. How do I find the radius from circumference?
- Use r = C / (2π). Divide the circumference by 6.2832. For C = 50, r = 50 / 6.2832 = 7.96 units. You can also use the Circ ↔ Radius tab in this circle calculator. Select "Circumference to Radius," type in the circumference, and press Calculate for an instant answer with the full working shown.
- 6. What is the inscribed angle theorem?
- The inscribed angle theorem states that an angle formed by two chords that meet at a point on the circle is half the central angle that subtends the same arc. So if the central angle is 120°, the inscribed angle is 60°. This theorem is tested in GCSE, IB, AP Pre-Calculus, and SAT geometry. Use the Theorems tab to solve inscribed angle problems step by step.
- 7. How do I calculate chord length?
- Use chord = 2r sin(θ/2), where θ is the central angle in radians. If your angle is in degrees, first convert: multiply by π/180. For r = 10 and θ = 60° (1.047 rad), chord = 2 × 10 × sin(0.524) = 20 × 0.5 = 10 units. The Arc & Sector tab automatically calculates and displays chord length alongside arc length and sector area.
- 8. What is the coin rotation paradox and how do I solve it?
- The coin rotation paradox asks how many times a small coin of radius r rotates as it rolls around the outside of a larger coin of radius R. The answer is R/r + 1, not R/r. The extra rotation happens because the small coin also revolves once around the large coin. For R = 3 and r = 1, the small coin rotates 4 times. Use the Theorems tab and select "Coin Rotation Paradox" to calculate and see the explanation.
- 9. Is this circle calculator useful for GCSE and IB exams?
- Yes. This tool covers every circle formula and theorem tested in GCSE Maths (both Foundation and Higher), IB SL and HL, and AP Precalculus. Use it to check your answers on past papers, understand where your working went wrong, and build speed before the exam. The Theorems tab covers the key GCSE circle theorems including inscribed angle, tangent-chord, and chord-chord properties.
- 10. Can I use this tool for both degrees and radians?
- Yes. The Arc & Sector tab has a dropdown to switch between Degrees and Radians. When you select Degrees, the calculator converts your angle to radians before applying L = rθ and A = ½r²θ, and it shows the conversion step in the output so you can see exactly how it is done. You can enter values like 90, 45, 120, or decimal radian values like 1.5708 directly.