The segment connecting the center of a circle and points on its circumference is called a radius. Moreover, in each circle all radii are equal to each other. The diameter of a circle is the straight line that connects two points on the circle and passes through its center. We will need all this to correctly calculate the area of ​​a circle. In addition, this value is calculated using the number Pi.

How to calculate the area of ​​a circle

For example, we have a circle with a radius of four centimeters. Let's calculate its area: S=(3.14)*4^2=(3.14)*16=50.24. Thus, the area of ​​the circle is 50.24 square centimeters.

Also, there is a special formula for calculating the area of ​​a circle through the diameter: S=(pi/4) d^2.

Let's look at an example of such a calculation of a circle through its diameter, knowing the radius of the figure. For example, we have a circle with a radius of four centimeters. First you need to find a diameter that is twice the radius itself: d=2R, d=2*4=8.

Now you should use the data obtained to calculate the area of ​​the circle using the formula described above: S=((3.14)/4 )*8^2=0.785*64=50.24.

As you can see, in the end we get the same answer as in the first case.

Knowledge of the standard formulas described above for correctly calculating the area of ​​a circle will help you easily find the missing values ​​and determine the area of ​​the sectors.

So, we know that the formula for calculating the area of ​​a circle is calculated by multiplying the constant value of Pi by the square of the radius of the circle itself. The radius itself can be expressed in terms of the actual circumference by substituting the expression in terms of the circumference into the formula. That is: R=l/2pi.

Now we need to substitute this equality into the formula for calculating the area of ​​a circle and as a result we get a formula for finding the area of ​​this geometric figure through the circumference: S=pi((l/2pi))^2=l^2/(4pi).

For example, we are given a circle whose circumference is eight centimeters. We substitute the value into the considered formula: S=(8^2)/(4*3.14)=64/(12.56)=5. And we get the area of ​​the circle equal to five square centimeters.

Circle calculator is a service specially designed for calculating the geometric dimensions of shapes online. Thanks to this service, you can easily determine any parameter of a figure based on a circle. For example: You know the volume of a ball, but you need to get its area. Nothing could be easier! Select the appropriate option, enter a numeric value, and click the Calculate button. The service not only displays the results of calculations, but also provides the formulas by which they were made. Using our service, you can easily calculate the radius, diameter, circumference (perimeter of a circle), area of ​​a circle and ball, and volume of a ball.

Calculate radius

The task of calculating the radius value is one of the most common. The reason for this is quite simple, because knowing this parameter, you can easily determine the value of any other parameter of a circle or ball. Our site is built exactly on this scheme. Regardless of what initial parameter you have chosen, the radius value is first calculated and all subsequent calculations are based on it. For greater accuracy of calculations, the site uses Pi, rounded to the 10th decimal place.

Calculate diameter

Calculating diameter is the simplest type of calculation that our calculator can perform. It is not at all difficult to obtain the diameter value manually; for this you do not need to resort to the Internet at all. The diameter is equal to the radius value multiplied by 2. Diameter is the most important parameter of a circle, which is extremely often used in everyday life. Absolutely everyone should be able to calculate and use it correctly. Using the capabilities of our website, you will calculate the diameter with great accuracy in a fraction of a second.

Find out the circumference

You can’t even imagine how many round objects there are around us and what an important role they play in our lives. The ability to calculate the circumference is necessary for everyone, from an ordinary driver to a leading design engineer. The formula for calculating the circumference is very simple: D=2Pr. The calculation can be easily done either on a piece of paper or using this online assistant. The advantage of the latter is that it illustrates all calculations with pictures. And on top of everything else, the second method is much faster.

Calculate the area of ​​a circle

The area of ​​a circle - like all the parameters listed in this article - is the basis of modern civilization. Being able to calculate and know the area of ​​a circle is useful for all segments of the population without exception. It is difficult to imagine a field of science and technology in which it would not be necessary to know the area of ​​a circle. The formula for calculation is again not difficult: S=PR 2. This formula and our online calculator will help you find out the area of ​​any circle without any extra effort. Our site guarantees high accuracy of calculations and their lightning-fast execution.

Calculate the area of ​​a sphere

The formula for calculating the area of ​​a ball is no more complicated than the formulas described in the previous paragraphs. S=4Pr 2 . This simple set of letters and numbers has been giving people the ability to fairly accurately calculate the area of ​​a ball for many years. Where can this be applied? Yes everywhere! For example, you know that the area of ​​the globe is 510,100,000 square kilometers. It is useless to list where knowledge of this formula can be applied. The scope of the formula for calculating the area of ​​a sphere is too wide.

Calculate the volume of the ball

To calculate the volume of the ball, use the formula V = 4/3 (Pr 3). It was used to create our online service. The website makes it possible to calculate the volume of a ball in a matter of seconds if you know any of the following parameters: radius, diameter, circumference, area of ​​a circle or area of ​​a ball. You can also use it for reverse calculations, for example, to know the volume of a ball and get the value of its radius or diameter. Thank you for taking a quick look at the capabilities of our circle calculator. We hope you liked our site and have already bookmarked the site.

Instructions

Use Pi to find the radius of a known area of ​​a circle. This constant sets the proportion between the diameter of a circle and the length of its border (circle). The length of a circle is the maximum area of ​​the plane that can be covered with its help, and the diameter is equal to two radii, therefore the area and radius also relate to each other with a proportion that can be expressed through the number Pi. This constant (π) is defined as the area (S) and the squared radius (r) of the circle. It follows from this that the radius can be expressed as the square root of the quotient of the area divided by Pi: r=√(S/π).

For a long time, Erastothenes headed the Library of Alexandria, the most famous library of the ancient world. In addition to calculating the size of our planet, he made a number of important inventions and discoveries. He invented a simple method for determining prime numbers, now called the “Sieve of Erasstophenes.”

He drew a “map of the world”, in which he showed all the parts of the world known to the ancient Greeks at that time. The map was considered one of the best for its time. He developed a system of longitude and latitude and a calendar that included leap years. Invented the armillary sphere, a mechanical device used by early astronomers to demonstrate and predict the apparent motion of stars in the sky. He also compiled a star catalog that included 675 stars.

Sources:

• The Greek scientist Eratosthenes of Cyrene was the first in the world to calculate the radius of the Earth
• Eratosthenes "Calculation of Earth"s Circumference
• Eratosthenes

Circles require a more careful approach and are much less common in tasks B5. At the same time, the general solution scheme is even simpler than in the case of polygons (see lesson “Areas of polygons on a coordinate grid”).

All that is required in such tasks is to find the radius of the circle R. Then you can calculate the area of ​​the circle using the formula S = πR 2. It also follows from this formula that to solve it it is enough to find R 2.

To find the indicated values, it is enough to indicate a point on the circle that lies at the intersection of the grid lines. And then use the Pythagorean theorem. Let's look at specific examples of calculating the radius:

Task. Find the radii of the three circles shown in the figure:

Let's perform additional constructions in each circle:

In each case, point B is chosen on the circle to lie at the intersection of the grid lines. Point C in circles 1 and 3 complete the figure to a right triangle. It remains to find the radii:

Consider triangle ABC in the first circle. According to the Pythagorean theorem: R 2 = AB 2 = AC 2 + BC 2 = 2 2 + 2 2 = 8.

For the second circle everything is obvious: R = AB = 2.

The third case is similar to the first. From triangle ABC using the Pythagorean theorem: R 2 = AB 2 = AC 2 + BC 2 = 1 2 + 2 2 = 5.

Now we know how to find the radius of a circle (or at least its square). Therefore, we can find the area. There are problems where you need to find the area of ​​a sector, and not the entire circle. In such cases, it is easy to find out what part of the circle this sector is, and thus find the area.

Task. Find the area S of the shaded sector. Please indicate S/π in your answer.

Obviously, the sector is one quarter of a circle. Therefore, S = 0.25 S circle.

It remains to find S of the circle - the area of ​​the circle. To do this, we perform an additional construction:

Triangle ABC is a right triangle. According to the Pythagorean theorem we have: R 2 = AB 2 = AC 2 + BC 2 = 2 2 + 2 2 = 8.

Now we find the area of ​​the circle and the sector: S circle = πR 2 = 8π ; S = 0.25 S circle = 2π.

Finally, the desired value is S /π = 2.

Sector area with unknown radius

This is a completely new type of task; there was nothing like it in 2010-2011. According to the condition, we are given a circle of a certain area (namely the area, not the radius!). Then, inside this circle, a sector is selected, the area of ​​which needs to be found.

The good news is that such problems are the easiest of all the area problems that appear in the Unified State Examination in mathematics. In addition, the circle and sector are always placed on a coordinate grid. Therefore, to learn how to solve such problems, just look at the picture:

Let the original circle have an area S circle = 80. Then it can be divided into two sectors of area S = 40 each (see step 2). Similarly, each of these “halves” sectors can be divided in half again - we get four sectors with area S = 20 each (see step 3). Finally, we can divide each of these sectors into two more - we get 8 “scraps” sectors. The area of ​​each of these “scraps” will be S = 10.

Please note: there is no finer division in any USE mathematics problem! Thus, the algorithm for solving Problem B-3 is as follows:

1. Cut the original circle into 8 “scraps” sectors. The area of ​​each of them is exactly 1/8 of the area of ​​the entire circle. For example, if according to the condition the circle has an area S of the circle = 240, then the “scraps” have an area S = 240: 8 = 30;
2. Find out how many “scraps” fit in the original sector, the area of ​​which needs to be found. For example, if our sector contains 3 “scraps” with an area of ​​30, then the area of ​​the desired sector is S = 3 · 30 = 90. This will be the answer.

That's it! The problem is solved practically orally. If something is still not clear, buy a pizza and cut it into 8 pieces. Each such piece will be the same sector-“scraps” that can be combined into larger pieces.

Now let’s look at examples from the trial Unified State Exam:

Task. A circle is drawn on checkered paper with an area of ​​40. Find the area of ​​the shaded figure.

So, the area of ​​the circle is 40. Divide it into 8 sectors - each with area S = 40: 5 = 8. We get:

Obviously, the shaded sector consists of exactly two “scraps” sectors. Therefore, its area is 2 · 5 = 10. That's the whole solution!

Task. A circle is drawn on checkered paper with an area of ​​64. Find the area of ​​the shaded figure.

Again, divide the entire circle into 8 equal sectors. Obviously, the area of ​​one of them is exactly what needs to be found. Therefore, its area is S = 64: 8 = 8.

Task. A circle is drawn on checkered paper with an area of ​​48. Find the area of ​​the shaded figure.

Again, divide the circle into 8 equal sectors. The area of ​​each of them is equal to S = 48: 8 = 6. The required sector contains exactly three sectors - “scraps” (see figure). Therefore, the area of ​​the required sector is 3 6 = 18.