Circle Properties
What is a Circle?
A circle is a two-dimensional shape consisting of all points in a plane that are a given distance from a given point, the center. The distance from the center to any point on the circle is called the radius ($r$).
- The Diameter ($d$) of a circle is any straight line segment that passes through the center and whose endpoints lie on the circle. It is twice the radius: $d = 2r$.
- The Circumference ($C$) is the distance around the circle. It is calculated by the formula $C = 2\pi r$ or $C = \pi d$.
Circles are fundamental geometric shapes with numerous applications in mathematics, science, and engineering.
Arc Length Calculator
The arc length ($s$) is the distance along a segment of the circumference of a circle.
Angle $\theta$ in radians: $s = r \theta$
Angle $\theta$ in degrees: $s = \frac{\theta}{360^\circ} \times 2\pi r$
Enter values and click calculate for a step-by-step solution.
Circle Area Calculator
The area ($A$) of a circle is the amount of two-dimensional space it occupies.
$A = \pi r^2$
Enter radius and click calculate for a step-by-step solution.
Sector Area Calculator
A sector of a circle is the region bounded by two radii and the intercepted arc, like a slice of pie.
Angle $\theta$ in radians: $A_{sector} = \frac{1}{2} r^2 \theta$
Angle $\theta$ in degrees: $A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2$
Enter values and click calculate for a step-by-step solution.
Equation of a Circle
The standard equation of a circle with center $(h, k)$ and radius $r$ is:
$(x-h)^2 + (y-k)^2 = r^2$
If the center is at the origin $(0,0)$, the equation simplifies to $x^2 + y^2 = r^2$. This equation is derived from the Pythagorean theorem, representing the distance from any point $(x,y)$ on the circle to its center $(h,k)$.
Circle Equation Plotter & Properties
Enter the center coordinates (h, k) and the radius (r) to plot the circle and see its equation.
Equation will appear here.
Radians & Degrees
Definition of a Radian
A radian is the standard unit of angular measure used in many areas of mathematics. An angle's measurement in radians is numerically equal to the length of a corresponding arc of a unit circle (a circle with radius 1).
Specifically, one radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.
Key Relationships:
- $2\pi \text{ radians} = 360^\circ$ (A full circle)
- $\pi \text{ radians} = 180^\circ$ (A half circle or straight angle)
- $1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57.296^\circ$
- $1^\circ = \frac{\pi}{180} \text{ radians} \approx 0.01745 \text{ radians}$
Radians & Degrees Converter
Degrees to Radians
Radians will appear here.
Radians to Degrees
Degrees will appear here.
Motion in a Circle
Introduction to Circular Motion
Circular motion occurs when an object moves along a circular path. Understanding this type of motion involves concepts like angular velocity, linear (tangential) velocity, centripetal acceleration, and centripetal force.
Linear & Angular Velocity Calculator
Angular Velocity ($\omega$): The rate at which an object rotates or revolves about an axis, or the rate at which the angular position ($\theta$) of an object changes over time ($t$). Measured in radians per second (rad/s) or degrees per second.
$\omega = \frac{\Delta\theta}{\Delta t}$
Linear (Tangential) Velocity ($v$): The rate at which an object covers distance along its circular path. It's the speed of the object tangent to the circle.
$v = r\omega$ (ensure $\omega$ is in rad/s for this formula)
Linear velocity (v) will appear here.
Centripetal Acceleration & Force Calculator
Centripetal Acceleration ($a_c$): The acceleration of an object moving in a circular path, directed towards the center of the circle. This acceleration is responsible for continuously changing the direction of the object's velocity, keeping it on the circular path.
$a_c = \frac{v^2}{r} = r\omega^2$
Centripetal Force ($F_c$): According to Newton's second law, a net force is required to cause acceleration. The centripetal force is this net force, always directed towards the center of the circular path, that maintains the circular motion.
$F_c = m a_c = \frac{mv^2}{r} = mr\omega^2$ (where $m$ is mass)
Results will appear here.