Introduction
This comprehensive guide presents all essential formulas related to circles, organized systematically for easy reference and learning. Whether you’re a student preparing for exams or an educator creating lesson plans, this resource provides accurate, complete formulas with clear explanations.
Basic Circle Properties
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Radius-Diameter Relationship | d = 2r | r = radius, d = diameter | Fundamental relationship between radius and diameter |
| Circumference | C = 2πr or C = πd | C = circumference, r = radius, d = diameter | Distance around the circle’s perimeter |
| Area | A = πr² | A = area, r = radius | Total area enclosed by the circle |
Arc and Central Angle Formulas
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Arc Length (Radians) | s = rθ | s = arc length, r = radius, θ = central angle (radians) | Length of arc when angle is in radians |
| Arc Length (Degrees) | s = (πrθ)/180° | s = arc length, r = radius, θ = central angle (degrees) | Length of arc when angle is in degrees |
| Central Angle (from Arc) | θ = s/r | θ = central angle (radians), s = arc length, r = radius | Finding central angle from arc length |
Sector Formulas
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Sector Area (Radians) | A = (1/2)r²θ | A = sector area, r = radius, θ = central angle (radians) | Area of sector when angle is in radians |
| Sector Area (Degrees) | A = (πr²θ)/360° | A = sector area, r = radius, θ = central angle (degrees) | Area of sector when angle is in degrees |
| Sector Perimeter | P = 2r + s | P = perimeter, r = radius, s = arc length | Total perimeter of sector including arc and two radii |
Chord Formulas
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Chord Length (Central Angle) | c = 2r sin(θ/2) | c = chord length, r = radius, θ = central angle | Length of chord given central angle |
| Chord Length (Perpendicular Distance) | c = 2√(r² – d²) | c = chord length, r = radius, d = perpendicular distance from center | Length of chord using perpendicular distance |
| Sagitta (Chord Height) | h = r – √(r² – (c/2)²) | h = sagitta, r = radius, c = chord length | Height of chord segment from chord to arc |
Segment Formulas
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Segment Area (Radians) | A = (1/2)r²(θ – sin θ) | A = segment area, r = radius, θ = central angle (radians) | Area between chord and arc |
| Segment Area (Degrees) | A = (πr²θ/360°) – (1/2)r² sin θ | A = segment area, r = radius, θ = central angle (degrees) | Area between chord and arc |
Circle Equations
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Standard Form | (x – h)² + (y – k)² = r² | (h,k) = center, r = radius | Standard equation with center at (h,k) |
| General Form | x² + y² + Dx + Ey + F = 0 | D, E, F = constants | General form of circle equation |
| Center from General Form | Center: (-D/2, -E/2) | D, E = coefficients from general form | Finding center from general equation |
| Radius from General Form | r = √[(D² + E² – 4F)/4] | D, E, F = coefficients from general form | Finding radius from general equation |
Parametric and Polar Forms
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Parametric Equations | x = h + r cos t, y = k + r sin t | (h,k) = center, r = radius, t = parameter | Parametric representation of circle |
| Polar Form (Center at Origin) | r = constant | r = radius (constant) | Circle centered at origin in polar coordinates |
Tangent and Secant Formulas
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Tangent Length (External Point) | t = √(d² – r²) | t = tangent length, d = distance from external point to center, r = radius | Length of tangent from external point |
| Power of a Point (External) | PT₁ × PT₂ = (d – r)(d + r) | PT₁, PT₂ = tangent segments, d = distance to center, r = radius | Power theorem for external point |
| Power of a Point (Secants) | PA × PB = PC × PD | PA, PB, PC, PD = segments of intersecting secants | Power theorem for secant lines |
Inscribed Angle and Arc Relationships
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Inscribed Angle Theorem | ∠inscribed = (1/2) × central angle | Inscribed angle = angle with vertex on circle | Inscribed angle is half the central angle |
| Angle Between Two Chords | ∠ = (1/2)(arc₁ + arc₂) | arc₁, arc₂ = intercepted arcs | Angle formed by intersecting chords |
| Angle Outside Circle | ∠ = (1/2) | arc₁ – arc₂ |
Advanced Relationships
| Formula Name | Formula | Variables | Description |
|---|---|---|---|
| Circle Area from Circumference | A = C²/(4π) | A = area, C = circumference | Finding area when only circumference is known |
| Radius from Area | r = √(A/π) | r = radius, A = area | Finding radius when only area is known |
| Arc Length as Fraction of Circumference | s = (θ/360°) × 2πr | s = arc length, θ = angle in degrees | Arc length as portion of full circumference |
Main Constants and Relationships
- π (Pi) ≈ 3.14159… (ratio of circumference to diameter)
- Radians to Degrees: multiply by 180°/π
- Degrees to Radians: multiply by π/180°
- Full Circle: 360° = 2π radians
Memory Tips for Students
- Circumference: “Pi × diameter” or “2 × pi × radius”
- Area: “Pi × radius squared” (πr²)
- Arc Length: Think of it as a fraction of the full circumference
- Sector Area: Think of it as a fraction of the full circle area
- Inscribed Angle: Always half the central angle subtending the same arc
Applications in Real Life
These formulas are essential for:
- Engineering: Designing circular components, gears, and wheels
- Architecture: Creating arches, domes, and circular structures
- Navigation: GPS calculations and great circle routes
- Physics: Rotational motion and wave calculations
- Computer Graphics: Rendering circles and curves
Frequently Asked Questions about Circle Formulas
Q. What is a circle in mathematics?
A circle is a closed two-dimensional shape where all points on the boundary are at an equal distance from a fixed point called the center.
Q. What are the basic formulas of a circle?
The most important circle formulas are:
- Circumference (C) = 2πr
- Area (A) = πr²
- Diameter (D) = 2r
Where r is the radius and π (pi) ≈ 3.14.
Q. What is the radius of a circle?
The radius is the distance from the center of the circle to any point on its boundary.
Q. What is the diameter of a circle?
The diameter is the distance across the circle passing through the center.
It is twice the radius.
Formula:
Diameter = 2 × Radius
Q. What is the circumference of a circle?
The circumference is the total distance around the boundary of a circle.
Formula:
Circumference = 2πr or πD
Q. What is the formula for the area of a circle?
The area of a circle represents the space enclosed inside it.
Formula:
Area = πr²
Q. Why is π (pi) used in circle formulas?
π is a mathematical constant that represents the ratio of a circle’s circumference to its diameter.
Its approximate value is 3.14 or 22/7.
Q. Can the area of a circle be calculated using the diameter?
Yes.
Formula:
Area = π × (D/2)²
Q. What are the units of area and circumference?
- Area is measured in square units (cm², m²).
- Circumference is measured in linear units (cm, m).
Q. What is the difference between a circle and a sphere?
- A circle is a 2D shape.
- A sphere is a 3D shape.
Circle formulas deal only with length and area, not volume.
Q. What is a semicircle and its formulas?
A semicircle is half of a circle.
- Area of semicircle = (πr²)/2
- Perimeter of semicircle = πr + 2r
Q. Are circle formulas important for exams?
Yes. Circle formulas are high-weightage topics in:
- CBSE & ICSE
- JEE, NEET foundation
- Olympiads
- Board exams
They frequently appear in numericals and word problems.
Q. How can students remember circle formulas easily?
A simple trick:
- Circle boundary → Circumference → 2πr
- Inside space → Area → πr²
Think “square for area” to remember r².
Q. What happens to the area if the radius is doubled?
If the radius is doubled, the area becomes four times.
Reason:
Area ∝ r²
Q. Where are circle formulas used in real life?
Circle formulas are used in:
- Architecture and construction
- Designing wheels, pipes, and plates
- Engineering and physics
- Sports fields and tracks
Circles are everywhere — math just explains them.