What is the formula for diameter. How to find and what will be the circumference of a circle?
Instructions
If only the diameter is known, the formula will look like “R = D/2”.
If length circle is unknown, but there is data on the length of a certain , then the formula will look like “R = (h^2*4 + L^2)/8*h”, where h is the height of the segment (is the distance from the middle of the chord to the most protruding part of the specified arc), and L is the length of the segment (which is not the length of the chord). A chord is a segment that connects two points circle.
note
It is necessary to distinguish between the concepts of “circle” and “circle”. A circle is part of a plane, which, in turn, is limited by a circle of a certain radius. To find the radius, you need to know the area of the circle. In this case, the equation will be “R = (S/π)^1/2”, where S is the area. To calculate the area, in turn, you need to know the radius (“S = πr^2”).
Knowing only the length diameter circles, you can calculate not only square circle, but also the area of some others geometric shapes. This follows from the fact that the diameters of circles inscribed or circumscribed around such figures coincide with the lengths of their sides or diagonals.
Instructions
If you need to find square(S) according to its known length diameter(D), multiply pi (π) by its length diameter, and divide the result by four: S=π ²*D²/4. For example, a circle is twenty centimeters, then its square can be calculated as follows: 3.14² * 20² / 4 = 9.86 * 400 / 4 = 986 centimeters.
If you need to find square square (S) along the diameter of the circle (D) around it, construct the length diameter squared, and divide the result in half: S=D²/2. For example, if the diameter of the circumscribed circle is twenty centimeters, then square square can be calculated as follows: 20² / 2 = 400 / 2 = 200 square centimeters.
If square square (S) must be found by the diameter of the circle inscribed in it (D), it is enough to construct the length diameter squared: S=D². For example, if the diameter of the inscribed circle is twenty centimeters, then square square can be calculated as follows: 20² = 400 square centimeters.
If you need to find square(S) according to known diameter m inscribed (d) and circumscribed (D) circles around it, then construct the length diameter inscribed circle into a square and divide by four, and to the result add half the product of the lengths of the inscribed and circumscribed circles: S=d²/4 + D*d/2. For example, if the diameter of the circumscribed circle is twenty centimeters, and the inscribed circle is ten centimeters, then square triangle can be calculated as follows: 10² / 4 + 20 * 10/2 = 25 + 100 = 125 square centimeters.
Use Google's built-in search engine to perform the necessary calculations. For example, so that using this search engine square right triangle according to the example from the fourth step, you need to enter the following search query: “10^2 / 4 + 20*10/2” and press the Enter key.
Sources:
- how to find the area of a circle by diameter
A circle is a flat geometric figure, all points of which are at the same and non-zero distance from a selected point, which is called the center of the circle. A straight line connecting any two points of a circle and passing through the center is called diameter. The total length of all boundaries of a two-dimensional figure, which is usually called the perimeter, is more often denoted in a circle as “ circumference" Knowing the circumference of a circle, you can calculate its diameter.
Instructions
To find the diameter, use one of the main properties of a circle, which is that the ratio of the length of its perimeter to the diameter is the same for absolutely all circles. Of course, constancy did not go unnoticed by mathematicians, and this proportion has long received its own - this is the number Pi (π is the first Greek word " circle" and "perimeter"). The numerical value of this is determined by the length of a circle whose diameter is equal to one.
Divide the known circumference of a circle by Pi to calculate its diameter. Since this number is “ ”, it does not have a finite value - it is a fraction. Round Pi according to the accuracy of the result you need to obtain.
Use some to calculate the length of the diameter if you can’t do it in your head. For example, you can use the one that is built into the Nigma or Google search engine - it is mathematical operations entered in “human” language. For example, if the known circumference is four meters, then to find the diameter you can “humanly” ask the search engine: “4 meters divided by pi.” But if you enter, for example, “4/pi” into the search query field, then the search engine will understand this formulation of the problem. In any case, the answer will be “1.27323954 meters”.
The question of the diameter of the globe is not as simple as it might seem at first glance, because the concept itself “ Earth"Very conditional. A real ball will always have the same diameter, no matter where a segment is drawn connecting two points on the surface of the sphere and passing through the center.
In relation to the Earth, it does not seem possible, since its spherical shape is far from ideal (in nature there are no ideal geometric figures and bodies at all; they are abstract geometric concepts). To accurately designate the Earth, scientists even had to introduce a special concept - “geoid”.
Official diameter of the Earth
The diameter of the Earth is determined by where it will be measured. For convenience, two indicators are taken as the officially recognized diameter: the diameter of the Earth at the equator and the distance between the North and South Poles. The first indicator is 12,756.274 km, and the second is 12,714, the difference between them is slightly less than 43 km.
These numbers do not make much of an impression; they are even inferior to the distance between Moscow and Krasnodar - two cities located in the same country. However, it was not easy to figure them out.
Calculating the diameter of the Earth
The diameter of the planet is calculated using the same geometric formula, like any other diameter.
To find the perimeter of a circle, you need to multiply its diameter by the number pi. Consequently, to find the diameter of the Earth, you need to measure its circumference in the appropriate section (along the equator or in the plane of the poles) and divide it by the number pi.
The first person to try to measure the circumference of the Earth was the ancient Greek scientist Eratosthenes of Cyrene. He noticed that in Siena (now Aswan) on the day of the summer solstice, the Sun was at its zenith, illuminating the bottom of a deep well. In Alexandria on that day it was 1/50 of the circle away from the zenith. From this, the scientist concluded that the distance from Alexandria to Syene is 1/50 of the circumference of the Earth. The distance between these cities is 5,000 Greek stadia (approximately 787.5 km), therefore the circumference of the Earth is 250,000 stadia (approximately 39,375 km).
Modern scientists have more advanced measuring instruments at their disposal, but they theoretical basis corresponds to the idea of Eratosthenes. At two points located several hundred kilometers from each other, the position of the Sun or certain stars in the sky is recorded and the difference between the results of the two measurements is calculated in degrees. Knowing the distance in kilometers, it is easy to calculate the length of one degree and then multiply it by 360.
To clarify the size of the Earth, both laser ranging and satellite systems observations.
Today it is believed that the circumference of the Earth at the equator is 40,075.017 km, and at the equator – 40,007.86. Eratosthenes was only slightly mistaken.
The size of both the circumference and diameter of the Earth is increasing due to meteorite matter that constantly falls on the Earth, but this process is very slow.
Sources:
- How the Earth was measured in 2019
Thus, the circumference ( C) can be calculated by multiplying the constant π per diameter ( D), or multiplying π by twice the radius, since the diameter is equal to two radii. Hence, circumference formula will look like this:
C = πD = 2πR
Where C- circumference, π - constant, D - circle diameter , R- radius of the circle.
Since the circle is the boundary circle, then the circumference can also be called the circumference of a circle or perimeter circle.
Circumference problems
Task 1. Find the circumference of a circle if its diameter is 5 cm.
Since the circumference is equal to π multiplied by the diameter, then the length of a circle with a diameter of 5 cm will be equal to:
C≈ 3.14 5 = 15.7 (cm)
Task 2. Find the length of a circle whose radius is 3.5 m.
First, find the diameter of the circle by multiplying the length of the radius by 2:
D= 3.5 2 = 7 (m)
Now let's find the circumference by multiplying π per diameter:
C≈ 3.14 7 = 21.98 (m)
Task 3. Find the radius of a circle whose length is 7.85 m.
To find the radius of a circle based on its length, you need to divide the circumference by 2 π
Area of a circle
The area of a circle is equal to the product of the number π per square radius. Formula for finding the area of a circle:
S = πr 2
Where S is the area of the circle, and r- radius of the circle.
Since the diameter of a circle is equal to twice the radius, then the radius equal to diameter, divided by 2:
Problems involving the area of a circle
Task 1. Find the area of a circle if its radius is 2 cm.
Since the area of a circle is π multiplied by the radius squared, then the area of a circle with a radius of 2 cm will be equal to:
S≈ 3.14 2 2 = 3.14 4 = 12.56 (cm 2)
Task 2. Find the area of a circle if its diameter is 7 cm.
First, find the radius of the circle by dividing its diameter by 2:
7:2=3.5(cm)
Now let's calculate the area of the circle using the formula:
S = πr 2 ≈ 3.14 3.5 2 = 3.14 12.25 = 38.465 (cm 2)
This task can be solved in another way. Instead of finding the radius first, you can use the formula for finding the area of a circle using the diameter:
| S = π | D 2 | ≈ 3,14 | 7 2 | = 3,14 | 49 | = | 153,86 | = 38.465 (cm 2) |
| 4 | 4 | 4 | 4 |
Task 3. Find the radius of the circle if its area is 12.56 m2.
To find the radius of a circle from its area, you need to divide the area of the circle π , and then extract from the obtained result Square root:
r = √S : π
therefore the radius will be equal to:
r≈ √12.56: 3.14 = √4 = 2 (m)
Number π
The circumference of objects surrounding us can be measured using a measuring tape or rope (thread), the length of which can then be measured separately. But in some cases, measuring the circumference is difficult or practically impossible, for example, the inner circumference of a bottle or simply the circumference of a circle drawn on paper. In such cases, you can calculate the circumference of a circle if you know the length of its diameter or radius.
To understand how this can be done, let’s take several round objects whose circumference and diameter can be measured. Let's calculate the ratio of length to diameter, and as a result we get the following series of numbers:
From this we can conclude that attitude the length of a circle to its diameter is a constant value for each individual circle and for all circles as a whole. This relationship is denoted by the letter π .
Using this knowledge, you can use the radius or diameter of a circle to find its length. For example, to calculate the length of a circle with a radius of 3 cm, you need to multiply the radius by 2 (this is how we get the diameter), and multiply the resulting diameter by π . As a result, using the number π We learned that the length of a circle with a radius of 3 cm is 18.84 cm.
Instructions
First you need the initial data for the task. The fact is that its condition cannot explicitly say what the radius is circle. Instead, the problem may give the length of the diameter circle. Diameter circle- a segment that connects two opposite points circle, passing through its center. Having analyzed the definitions circle, we can say that the length of the diameter is twice the length of the radius.
Now we can accept the radius circle equal to R. Then for the length circle you need to use the formula:
L = 2πR = πD, where L is the length circle, D - diameter circle, which is always 2 times the radius.
note
A circle can be inscribed in a polygon or described around it. Moreover, if the circle is inscribed, then at the points of contact with the sides of the polygon it will divide them in half. To find out the radius of the inscribed circle, you need to divide the area of the polygon by half its perimeter:
R = S/p.
If a circle is circumscribed around a triangle, then its radius is found using the following formula:
R = a*b*c/4S, where a, b, c are the sides of a given triangle, S is the area of the triangle around which the circle is circumscribed.
If you want to describe a circle around a quadrilateral, this can be done if two conditions are met:
The quadrilateral must be convex.
The sum of the opposite angles of the quadrilateral should be 180°
In addition to the traditional caliper, stencils can also be used to draw a circle. Modern stencils include a circle different diameters. These stencils can be purchased at any office supply store.
Sources:
- How to find the circumference of a circle?
A circle is a closed curved line, all points of which are at equal distances from one point. This point is the center of the circle, and the segment between the point on the curve and its center is called the radius of the circle.
Instructions
If a straight line is drawn through the center of a circle, then its segment between two points of intersection of this line with the circle is called the diameter of the given circle. Half the diameter, from the center to the point where the diameter intersects the circle is the radius
circles. If a circle is cut at an arbitrary point, straightened and measured, then the resulting value is the length of the given circle.
Draw several circles with different compass solutions. Visual comparison suggests that a larger diameter outlines a larger circle bounded by a circle with a larger length. Therefore, between the diameter of a circle and its length there is a direct relationship proportional dependence.
In its physical meaning, the “circumference length” parameter corresponds to a bounded by a broken line. If we inscribe a regular n-gon with side b into a circle, then the perimeter of such a figure P is equal to the product of side b by the number of sides n: P=b*n. Side b can be determined by the formula: b=2R*Sin (π/n), where R is the radius of the circle into which the n-gon is inscribed.
As the number of sides increases, the perimeter of the inscribed polygon will increasingly approach L. Р= b*n=2n*R*Sin (π/n)=n*D*Sin (π/n). The relationship between the circumference L and its diameter D is constant. The ratio L/D=n*Sin (π/n) as the number of sides of an inscribed polygon tends to infinity tends to the number π, a constant value called “pi” and expressed as infinite decimal. For calculations without application computer technology the value π=3.14 is accepted. The circumference of a circle and its diameter are related by the formula: L= πD. For a circle, divide its length by π=3.14.