Radian Measure Of Central Angle

Accept y'all ever wondered how to find the central angle of a circle? The central angle calculator is hither to help; the simply variables you need are the arc length and the radius.

Read on to learn the definition of a central bending and how to utilize the central bending formula.

What is a central bending?

A cardinal bending is an angle with a vertex at the centre of a circle whose artillery extend to the circumference. Y'all tin imagine the cardinal angle existence at the tip of a pizza slice in a large round pizza.

Yous can find the central angle of a circle using the formula:

θ = L / r

where θ is the central angle in radians, 50 is the arc length and r is the radius.

Where does the central bending formula come from?

The simplicity of the central angle formula originates from the definition of a radian. A radian is a unit of angular size, where i radian is defined equally a central angle (θ) whose arc length is equal to the radius (L = r).

The circle bending calculator in terms of pizza

Considering maths can make people hungry, we might better empathize the central angle in terms of pizza. What would the central angle be for a slice of pizza if the crust length (L) was equal to the radius (r)?

Since the trouble defines L = r, and we know that ane radian is defined as the central angle when L = r, we tin can see that the cardinal angle is 1 radian. We could also use the key bending formula equally follows:

θ = Fifty / r

= L / L

= 1 rad

How many pizza slices with a central angle of 1 radian could you cut from a round pizza?

In a complete circular pizza, nosotros know that the cardinal angles of all the slices volition add up to 2π radians = 360°. Since each slice has a cardinal bending of ane radian, nosotros volition demand 2π / 1 = 2π slices, or six.28 slices to make full up a consummate circle.

We go far at the same answer if we remember this problem in terms of the pizza crust: we know that the circumference of a circumvolve is 2πr. Since the chaff length = radius, then 2πr / r = 2π crusts will fit along the pizza perimeter.

At present, if y'all are all the same hungry, accept a look at the sector area computer to summate the area of each pizza slice!

Bonus challenge - How far does the Earth travel in each flavor?

Try using the central angle calculator in opposite to assistance solve this problem. The World is approximately 149.6 million km abroad from the Sun. If the Earth travels about 1 quarter of its orbit each season, how many km does the Earth travel each season (eastward.g., from spring to summertime)?

Let's arroyo this problem pace-past-stride:

Simplify the problem by bold the Earth'south orbit is circular (The Earth'southward orbit is actually elliptical, and constantly irresolute). In this model, the Dominicus is at the centre of the circle, and the World's orbit is the circumference.
The radius is the altitude from the Globe and the Sunday: 149.vi meg km.
The central angle is a quarter of a circumvolve: 360° / 4 = 90°.
Use the central angle calculator to find arc length.

You can try the terminal adding yourself by rearranging the formula equally:

L = θ * r

Then convert the central angle into radians: 90° = one.57 rad, and solve the equation:

L = ane.57 * 149.half dozen million km

L = 234.9 million km.

When we assume that for a perfectly circular orbit, the Earth travels approximately 234.ix one thousand thousand km each flavour!

FAQ

How to find the central angle of a circumvolve?

To find the cardinal angle of a circle, use the formula:
θ = L / r
where:
θ is the central angle in radians;
Fifty is the arc length; and
r is the radius.

To find the key angle of a circle, you need to calculate the ratio of arc length to the radius of a circle.
You tin can imagine the central angle beingness at the tip of a pizza slice in a big circular pizza.