What on Earth is a central angle? It sounds like something you’d learn in a high school geometry class, and chances are, you did. But it’s useful to know that this concept of math has real-world applications, too! While the formula for finding the measure of a central angle might seem intimidating at first, we’ll break down each part step-by-step so you can learn what the heck it’s all about. Before we get into the nitty-gritty details, though, let’s start with a definition.

What is a central angle?

A central angle is the angle formed by two radii of a circle. It can be any size you want, so long as the radius stays constant.

central angle with radius

If you have a radius, or diameter (if it’s an unbroken line), draw an arc that goes from one end of that line to the other and then measure off 90 degrees on either side of this arc. This will give you two points where your central angle touches your radius line; these are called vertices for obvious reasons!

Where does the central angle formula come from?

Why is it called the central angle formula? Well, it comes from a trigonometric identity called “the law of cosines,” which means that if you have three sides of a triangle and the angle between them, you can use this identity to figure out what the fourth side would be.

The law of cosines is derived from Pythagorean Theorem, where we know that

a^2 + b^2 = c^2 \text { (for right triangles)}

Central angle formula

The central angle formula is very simple: the sum of the two angles of a central angle equals 180 degrees.

This means that if you have an angle, then you can use this formula to find its opposite side. In other words, if you know one side and have an idea about what its length should be, then this is how you can go about finding it!

We’ll show you how to use this formula with a few examples below.

How to find the central angle of a circle?

In order to find the central angle of a circle, you have to follow a few simple steps:

  • Find the radius of the circle. The radius is a line segment that extends from any point on a circle to its center.
  • Calculate the angle between the radius and another line within a circle, called an arc. This means you’ll have to calculate it in three dimensions instead of just two if you’re dealing with a sphere or another figure where there’s no defined center point.
  • Take this number, add 90 degrees (since we’re measuring angles in terms of degrees) and then subtract anything between 360 and 359 if necessary‚ÄĒthis will give us our central angle!

Central angle formula explanation

The central angle formula is pretty simple:

CA = \frac {S}{r}

S is the length of the arc, and r is the radius of the circle.

The central angle at the vertex of a given triangle is equal to the sum of all its interior angles minus 2 times the measure of one exterior side (opposite vertex).

In other words, if you know the measures of two sides and an interior angle, then you can find out how big your third side must be. This will help you solve problems like “Find me my missing side.”

How to use the central angle calculator

Our central angle calculator uses the formula that we mentioned above. So, in order to calculate the central angle using this calculator, all you need to do, is enter the arc length and the radius of the circle (in any measurements you want), and you will get the central angle.

The central angle can be calculated in any measurements you want, from standard degrees or radians, to gradians, turns, minutes of arc, and more.

Central angle example

The central angle in a circle is the angle at the center of the circle, formed by two radii intersecting at right angles. The central angle in a square, equilateral triangle or rectangle is a right angle. A right angle has an inscribed arc that forms 1/8th of its circumference and lies on its circumference.

FAQ

What is the central angle?

A central angle is the angle formed by two radii of a circle.

How to find central angle with arc length?

You can calculate it using the formula we mentioned above: CA = S/r, where S is the arc length and r is the radius of the circle.

What is an arc length?

The arc length is the distance between the two points on the outer layer of the circle that form the central angle.