Central Angle Calculator
Calculate central angles, arc lengths, and radius of circles using the formula θ = L/r
Calculate Central Angle Properties
The length of the arc along the circle's circumference
The distance from the center to the circumference
⚠️ Please enter a positive radius to calculate the central angle.
Example: Pizza Slice Calculation
Problem
Given: A circular pizza with radius = 10 inches, arc length (crust) = 8 inches
Find: What is the central angle of this pizza slice?
Solution
Step 1: Use the central angle formula: θ = L / r
Step 2: Substitute values: θ = 8 / 10 = 0.8 radians
Step 3: Convert to degrees: 0.8 × (180°/π) = 45.84°
Answer: The central angle is 0.8 radians or 45.84°
Central Angle Formulas
Central Angle Facts
A central angle has its vertex at the circle's center
Think of it as the angle at the tip of a pizza slice
Full circle = 360° = 2π radians
1 radian ≈ 57.3 degrees
When arc length = radius, angle = 1 radian
Real-World Applications
Earth's orbital mechanics and seasonal calculations
Architecture and construction of curved structures
Mechanical engineering and gear calculations
Art and design with circular patterns
Understanding Central Angles
What is a Central Angle?
A central angle is an angle whose vertex is at the center of a circle and whose sides are radii that extend to the circumference. Think of it as the angle at the tip of a pizza slice in a circular pizza.
The Radian Definition
The elegance of the central angle formula θ = L/r comes from the definition of a radian. One radian is defined as the central angle where the arc length equals the radius (L = r). This makes the calculation straightforward and universal.
Key Relationships
- •When L = r, then θ = 1 radian ≈ 57.3°
- •Full circle: θ = 2π radians = 360°
- •Semicircle: θ = π radians = 180°
- •Quarter circle: θ = π/2 radians = 90°
Formula Derivations
From basic proportions:
Arc length / Circumference = Angle / Full circle
L / (2πr) = θ / (2π)
L = θ × r
θ = L / r
Earth's Seasonal Journey
Problem: How far does Earth travel in one season?
Given: Earth-Sun distance = 149.6 million km
Solution: θ = 90° = π/2 radians
Arc length: L = (π/2) × 149.6M = 234.9M km
Navigation & GPS
Central angles help calculate distances on Earth's surface and determine shortest paths between geographic locations.
Engineering Design
Used in designing curved roads, bridges, and mechanical components like gears, pulleys, and rotating machinery.
Computer Graphics
Essential for creating smooth curves, animations, and 3D rotations in video games and digital art applications.