Circle is the main figure in geometry, the properties of which are considered at school in the 8th grade. One of the typical problems associated with a circle is to find the area of some part of it, which is called a circular sector. The article provides formulas for the area of a sector and the length of its arc, as well as an example of their use for solving a specific problem.
The concept of a circle and a circle
Before giving the formula for the area of a sector of a circle, let's consider what the indicated figure is. According to the mathematical definition, a circle is understood as such a figure on a plane, all points of which are equidistant from some one point (center).
When considering a circle, the following terminology is used:
- Radius - a segment that is drawn from the central point to the curve of the circle. It is usually denoted by the letter R.
- Diameter is a segment that connects two points of the circle, but also passes through the center of the figure. It is usually denoted by the letter D.
- Arc is part of a curved circle. It is measured either in units of length or using angles.
Circle is another important geometry figure, it is a collection of points that is bounded by a circle curve.
Circle area and circumference
The values noted in the title of the item are calculated using two simple formulas. They are listed below:
- Circumference: L=2piR.
- Area of a circle: S=piR2.
In these formulas, pi is some constant called Pi. It is irrational, that is, it cannot be expressed exactly as a simple fraction. Pi is approximately 3.1416.
As you can see from the above expressions, in order to calculate the area and length, it is enough to know only the radius of the circle.
The area of the sector of the circle and the length of its arc
Before considering the corresponding formulas, we recall that the angle in geometry is usually expressed in two main ways:
- in sexagesimal degrees, and a full rotation around its axis is 360o;
- in radians, expressed as fractions of pi and related to degrees by the following equation: 2pi=360o.
A circle sector is a figure bounded by three lines: an arc of a circle and two radii located at the ends of this arc. An example of a circular sector is shown in the photo below.
Getting an idea of what a sector for a circle is, it's easyunderstand how to calculate its area and the length of the corresponding arc. It can be seen from the figure above that the arc of the sector corresponds to the angle θ. We know that a full circle corresponds to 2pi radians, so the formula for the area of a circular sector will take the form: S1=Sθ/(2pi)=piR 2θ/(2pi)=θR2/2. Here the angle θ is expressed in radians. A similar formula for the sector area, if the angle θ is measured in degrees, will look like this: S1=piθR2/360.
The length of the arc forming a sector is calculated by the formula: L1=θ2piR/(2pi)=θR. And if θ is known in degrees, then: L1=piθR/180.
Example of problem solving
Let's use the example of a simple problem to show how to use the formulas for the area of a sector of a circle and the length of its arc.
It is known that the wheel has 12 spokes. When the wheel makes one complete revolution, it covers a distance of 1.5 meters. What is the area enclosed between two adjacent spokes of the wheel, and what is the length of the arc between them?
As you can see from the corresponding formulas, in order to use them, you need to know two quantities: the radius of the circle and the angle of the arc. The radius can be calculated from knowing the circumference of the wheel, since the distance traveled by it in one revolution corresponds exactly to it. We have: 2Rpi=1.5, whence: R=1.5/(2pi)=0.2387 meters. The angle between the nearest spokes can be determined by knowing their number. Assuming that all 12 spokes divide the circle evenly into equal sectors, we get 12 identical sectors. Accordingly, the angular measure of the arc between the two spokes is: θ=2pi/12=pi/6=0.5236 radian.
We have found all the necessary values, now they can be substituted into the formulas and calculate the values required by the condition of the problem. We get: S1=0.5236(0.2387)2/2=0.0149 m2, or 149cm2; L1=0.52360.2387=0.125 m or 12.5 cm.
