A circle and a circle are two perfect flat figures, the properties of which are studied without fail in any school geometry course. In this article, we will look at how to find the circumference and area of a circle using simple mathematical formulas.
What is the difference between a circle and a circle?
Before proceeding to the consideration of the formulas for the circumference and the area of the circle, the definitions of these figures should be given.
In geometry, a circle is understood to mean a collection of points on a plane that are located at the same distance from a fixed point R. In turn, a circle is a set of points on a plane that are located at distances equal to or less than a certain number R from a given point. In other words , the circle represents one single curved line, and the circle occupies a certain area. A circle is the "rim" of a circle.
That is why, the question of how to find the area of a circle is considered to be incorrectly posed. The circle (single line) has no area, but it has a length. For the circle, talking about the area makes sense, in addition, you can also talk about the length of the circle that limits it.
The main properties of the figures in question
The circle and circle have a number of common characteristics, which are briefly listed below:
- They have a radius R (the length of the segment connecting the center of the figure with its edge).
- If we draw a segment through them that passes through the center and connects the two edges of the figure, then it will be equal to 2 * R and is called the diameter (D).
- Any axis passing through the diameter divides the figure into two equal parts.
- Turning at an arbitrary angle of a circle or circle around an axis passing through his / her center and perpendicular to the plane of the figure is a symmetry operation.
Formulas for circumference and circle area
Having become acquainted with the concept and basic properties of the considered flat figures, we can proceed to the quantitative determination of their sizes. The circumference and circle area are calculated using the following two formulas:
1. L = 2 * π * R.
2. S = π * R²
From these formulas it follows that the value of R - radius - fully characterizes the properties of both figures. The value of L is measured in meters (~ R), and S - in square meters (~ R²).
In the formulas, the symbol π represents some constant, which is an irrational number (it cannot be calculated exactly). With an accuracy of 4 decimal places, the number π is 3.1416. Note that when performing calculations, this constant can be replaced by the fraction 201/64. If you calculate the value of this fraction, you get the number 3.1406, which is only 0.03% different from the true constant.
Note that the formula for the circumference is also valid for determining a similar characteristic of the circle.
These formulas can be rewritten through the diameter, given that D = 2 * R, we obtain:
1) L = π * D;
2) S = π * D² / 4.
Using the considered formulas to solve the problem
We use formulas for the area of a circle and circumference for solving problems. For example, Masha has a piece of rectangular-shaped fabric, the dimensions of which are 5 x 4 meters. It is necessary to determine what maximum size the circle can cut from this fabric.
The meaning of this task is to determine the size of the circle inscribed in the quadrangle. This situation is depicted in the figure below.
From the figure you can see that the diameter of the inscribed circle D will be equal to the length of the smallest side of the quadrangle, in this case D = 4 meters. Knowing the diameter, you can directly apply the formulas that are written for the length and area of this figure in the previous paragraph of the article. We have:
1. L = π * D = 3.1416 * 4 = 12.5664 m.
2. S = π * D² / 4 = 3.1416 * 4² / 4 = 12.5664 m².
We got an interesting result: the area of a circle is exactly the same as the length of its circle, but the units of measurement are different for them. This result is a simple coincidence, since D = 4 is the only number for which the absolute values of L and S are equal.