A central angle subtended by a significant arc has a measure greater than 180 degrees. To find the length of an arc in a circle, use the arc length formula: l = rth, l = r th, where th is in radians. A = 12th r2 and A = 1 2 th r2 in the sector area, where th is in radians. Thus, arclength = 12th r2 sin(1/2 th) = 3.1416 r2 sin(1/2 th). You can calculate sin(1/2 th) in terms of th using the sine rule. Here, 1/2 th is between 0 and pi/2. Thus, sin(1/2 th) = sin(pi/2 - 1/2 th) = -sin(1/2 th), so that l = 3.1416 r2 -sin(1/2 th).

In general, if x is a real number such that 0 <= x < 1, then sin x is defined and lies in the range -1 to +1. It can be shown **that sin** x is positive if x is less than 45 degrees or negative if x is greater than 90 degrees. This means that the length of an arc is always greater than zero and less than 4r2. If you need **an arc length** that exceeds 4r2, use one of the forms for infinite lengths instead.

You will also learn the sector area equation. An arc's length is determined by the radius of a circle and the center angle. We know that the arc length is equal to the circumference at an angle of 360 degrees (2p). As a result, because the proportion between angle and arc length remains constant, we may say: angle = circumference / sector area.

Thus, **the sector area** can be calculated by dividing the circumference by the angle. The division is done using the rule that says that if you have two quantities that are in ratio A to B, then the total is just A times B. In **this case**, the total is the circumference divided by the angle. That's how we can calculate the sector area.

The formula looks like this: sector area = circumference / angle. Or, put **another way**, sector area = 2πr where π stands for pi (3.14...), r is the radius of the circle and angle is measured in degrees.

As you can see, the sector area is simply the ratio between the circumference and the angle. This means that the larger the angle, the smaller the sector area. On the other hand, the smaller the angle, the larger the sector area.

For example, if we want to find out **what kind** of **sector area** we get when the angle is 30 degrees, we can use the formula above and enter the numbers directly into it: sector area = 32π.

The length of the arc is a component of the circle's circumference; the measure of the arc is the measure of **the central angle**. If you know one, you know them both. However, they are not the same thing! The length of an arc is always greater than or equal to the measure of its central angle, but that isn't always true of the measures of arcs on a circle. For example, if an arc were twice as long as its central angle, it would be half as wide. This would make its measure equal to $90^\circ$, which is not possible.

An arc is said to have a length of 180 degrees when its central angle is exactly 2$\pi$ radians. Because a full rotation equals 2$\pi$ radians, this means that the arc covers the whole circle. An arc of less than 180 degrees has a length less than 2$\pi$ radians. These arcs cover **only a part** of the circle. Any arc can have a length greater than 2$\pi$ radians, although these cases are rare.

It may help to think about it this way: The length of an arc is the distance along the circle from **one endpoint** to the other.

Sector Area and Arc Length The area of a sector may also be calculated using its radius and arc length. The area, A, of a circle with a radius, r, and an arc length, L, is given by the formula: A = (r x L) 2. As an example, if the radius of the circle is 10 meters and the arc length is 5 meters, then the area is 50 square meters.

The formula for calculating the area of an ellipse uses **the same variables** as the formula for a circle but adds the additional variable, E, which represents the ratio of the lengths of the two major axes of the ellipse. If the major axes are a and b respectively, then E = a/b. The area of an ellipse can be calculated using this formula: A = (a x b) × π × r 2. For our example, the area is 42.45 square meters.

An angle is said to be rectilinear if it can be divided into **two straight lines** without any gaps in between. Angles that cannot be so divided are called non-rectilinear. An arc is defined as the interior of a circle with some part of the circle's boundary. Thus, an arc is a portion of a circle bounded by two radii from the center. The term "arc" comes from **a Latin word** meaning "portion". Thus, an arc is a portion of a circle.

Arc length = 2pr (th/360) th = the angle (in degrees) subtended by an arc at the circle's center. Here, "r" is the radius of the circle.

Example: If an arc extends from the center of a circle with a radius of 5 units, then it makes an angle of 45 degrees with the horizontal. Therefore, the length of the arc is 10 units.

This relationship can be used to find the area of a circle or the value of an arc segment. It also can be applied to solve problems in geometry. For example, if the radius of a circle is 30 miles and an arc covers a distance of **12 miles**, how far does the circle cover? The answer is that the circle covers **20,480 feet** or 0.01 miles.

Here is another problem that can be solved using this relationship: What is the average speed of a car that travels on **a circular road** whose radius is five miles? The answer is that the car travels a distance of four miles with each rotation of the wheel or else it passes **every point** on the circle twice. Thus, its average speed is two miles per hour.

As you can see, this relationship comes in very handy for solving **many problems** in math and physics.