The circle is the main figure in geometry, the properties of which are considered in school in the 8th grade. One of the typical problems associated with a circle is to find the area of some of its part, which is called the circular sector. The article provides formulas for the area of the sector and the length of its arc, as well as an example of their use for solving a specific problem.
The concept of circle and circle
Before we give the formula for the area of the sector of the circle, consider what this figure represents. According to the mathematical definition, a circle is understood to mean such a figure on a plane, all points of which are equidistant from a single point (center).
When considering a circle, they use the following terminology:
- Radius - a segment that is drawn from the center point to the circle curve. It is customary to denote the letter R.
- Diameter is a segment that connects two points of a circle, but also passes through the center of the figure. It is usually denoted by the letter D.
- An arc is part of a circle curve. Measure it either in units of length, or using angles.
A circle is another important figure in geometry; it is a collection of points that is bounded by a circle curve.
Circle area and circumference
The values indicated in the item name are calculated using two simple formulas. They are listed below:
- Circumference: L = 2 * pi * R.
- Circle area: S = pi * R 2 .
In these formulas, pi is a constant called the Pi number. It is irrational, that is, it cannot be accurately expressed by a simple fraction. Approximately Pi is 3.1416.
As can be seen from the above expressions, in order to calculate the area and length, it is enough to know only the radius of the circle.
The sector area of a circle and the length of its arc
Before considering the appropriate formulas, we recall that the angle in geometry is usually expressed in two main ways:
- in six-decimal degrees, and a full revolution around its axis is 360 o ;
- in radians, which are expressed in fractions of the number pi and are related to degrees by the following equality: 2 * pi = 360 o .
The 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.
Having an idea of what a sector is for a circle, it is easy to understand how to calculate its area and the length of the corresponding arc. The figure above shows that the arc of the sector corresponds to the angle θ. We know that the full circle corresponds to 2 * pi radians, which means that the formula for the area of the circular sector takes the form: S 1 = S * θ / (2 * pi) = pi * R 2 * θ / (2 * pi) = θ * R 2/2 . Here, the angle θ is expressed in radians. A similar formula for the sector area if the angle θ is measured in degrees, will be: S 1 = pi * θ * R 2/360.
The length of the arc forming the sector is calculated by the formula: L 1 = θ * 2 * pi * R / (2 * pi) = θ * R. And if θ is known in degrees, then: L 1 = pi * θ * R / 180.
Problem solving example
Let us show by the example of a simple problem how to use the formulas of the area of a sector of a circle and the length of its arc.
The wheel is known to have 12 spokes. When the wheel makes one full revolution, it travels 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 can be seen from the corresponding formulas, in order to use them, it is necessary to know two quantities: the radius of the circle and the angle of the arc. The radius can be calculated based on the knowledge of the circumference of the wheel, since the distance traveled by it in one revolution exactly corresponds to it. We have: 2 * R * pi = 1.5, whence: R = 1.5 / (2 * pi) = 0.2387 meters. The angle between the nearest knitting needles 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 = 2 * pi / 12 = pi / 6 = 0.5236 radians.
We found all the necessary quantities, now they can be substituted into the formulas and the values required by the condition of the problem can be calculated. We get: S 1 = 0.5236 * (0.2387) 2/2 = 0.0149 m 2, or 149 cm 2 ; L 1 = 0.5236 * 0.2387 = 0.125 m, or 12.5 cm.