What is the area of ​​a circle? Formula. Area of ​​a Circle in Problem B5

How to find the area of ​​a circle? First find the radius. Learn to solve simple and complex problems.

A circle is a closed curve. Any point on the circle line will be the same distance from the center point. A circle is a flat figure, so solving problems involving finding area is easy. In this article we will look at how to find the area of ​​a circle inscribed in a triangle, trapezoid, square, and circumscribed around these figures.

To find the area of ​​a given figure, you need to know what the radius, diameter and number π are.

Radius R is the distance limited by the center of the circle. The lengths of all R-radii of one circle will be equal.

Diameter D is a line between any two points on a circle that passes through the center point. The length of this segment is equal to the length of the R-radius multiplied by 2.

Number π is a constant value that is equal to 3.1415926. In mathematics, this number is usually rounded to 3.14.

Formula for finding the area of ​​a circle using the radius:



Examples of solving problems on finding the S-area of ​​a circle using the R-radius:

Task: Find the area of ​​a circle if its radius is 7 cm.

Solution: S=πR², S=3.14*7², S=3.14*49=153.86 cm².

Answer: The area of ​​the circle is 153.86 cm².

The formula for finding the S-area of ​​a circle through the D-diameter:

Examples of solving problems to find S if D is known:

————————————————————————————————————————-

Task: Find the S of a circle if its D is 10 cm.

Solution: P=π*d²/4, P=3.14*10²/4=3.14*100/4=314/4=78.5 cm².

Answer: The area of ​​a flat circular figure is 78.5 cm².

Finding S of a circle if the circumference is known:

First we find what the radius is equal to. The circumference of the circle is calculated by the formula: L=2πR, respectively, the radius R will be equal to L/2π. Now we find the area of ​​the circle using the formula through R.

Let's consider the solution using an example problem:

———————————————————————————————————————-

Task: Find the area of ​​a circle if the circumference L is known - 12 cm.

Solution: First we find the radius: R=L/2π=12/2*3.14=12/6.28=1.91.

Now we find the area through the radius: S=πR²=3.14*1.91²=3.14*3.65=11.46 cm².

Answer: The area of ​​the circle is 11.46 cm².



Finding the area of ​​a circle inscribed in a square is easy. The side of a square is the diameter of a circle. To find the radius, you need to divide the side by 2.

Formula for finding the area of ​​a circle inscribed in a square:

Examples of solving problems of finding the area of ​​a circle inscribed in a square:

———————————————————————————————————————

Task #1: The side of a square figure is known, which is 6 centimeters. Find the S-area of ​​the inscribed circle.

Solution: S=π(a/2)²=3.14(6/2)²=3.14*9=28.26 cm².

Answer: The area of ​​a flat circular figure is 28.26 cm².

————————————————————————————————————————

Task No. 2: Find S of a circle inscribed in a square figure and its radius if one side is a=4 cm.

Decide this way: First we find R=a/2=4/2=2 cm.

Now let's find the area of ​​the circle S=3.14*2²=3.14*4=12.56 cm².

Answer: The area of ​​a flat circular figure is 12.56 cm².



It is a little more difficult to find the area of ​​a circular figure described around a square. But, knowing the formula, you can quickly calculate this value.

The formula for finding S a circle circumscribed about a square figure:

Examples of solving problems on finding the area of ​​a circle circumscribed around a square figure:

Task





A circle that is inscribed in a triangular figure is a circle that touches all three sides of the triangle. You can fit a circle into any triangular figure, but only one. The center of the circle will be the intersection point of the bisectors of the angles of the triangle.

The formula for finding the area of ​​a circle inscribed in an isosceles triangle:



Once the radius is known, the area can be calculated using the formula: S=πR².

Formula for finding the area of ​​a circle inscribed in a right triangle:



Examples of problem solving:

Task No. 1



If in this problem you also need to find the area of ​​a circle with a radius of 4 cm, then this can be done using the formula: S=πR²

Task No. 2



Solution:



Now that the radius is known, we can find the area of ​​the circle using the radius. See the formula above in the text.

Task No. 3



Area of ​​a circle circumscribed about a right and isosceles triangle: formula, examples of problem solving

All formulas for finding the area of ​​a circle boil down to the fact that you first need to find its radius. When the radius is known, then finding the area is simple, as described above.

The area of ​​a circle circumscribed about a right and isosceles triangle is found by the following formula:



Examples of problem solving:



Here is another example of solving a problem using Heron's formula.



Solving such problems is difficult, but they can be mastered if you know all the formulas. Students solve such problems in 9th grade.

Area of ​​a circle inscribed in a rectangular and isosceles trapezoid: formula, examples of problem solving

An isosceles trapezoid has two equal sides. A rectangular trapezoid has one angle equal to 90º. Let's look at how to find the area of ​​a circle inscribed in a rectangular and isosceles trapezoid using the example of solving problems.

For example, a circle is inscribed in an isosceles trapezoid, which at the point of contact divides one side into segments m and n.

To solve this problem you need to use the following formulas:



Finding the area of ​​a circle inscribed in rectangular trapezoid, is produced according to the following formula:



If known side, then you can find the radius through this value. The height of the side of a trapezoid is equal to the diameter of the circle, and the radius is half the diameter. Accordingly, the radius is R=d/2.

Examples of problem solving:



A trapezoid can be inscribed in a circle when the sum of its opposite angles is 180º. Therefore, you can only inscribe an isosceles trapezoid. The radius for calculating the area of ​​a circle circumscribed about a rectangular or isosceles trapezoid is calculated using the following formulas:





Examples of problem solving:



Solution: The large base in this case passes through the center, since an isosceles trapezoid is inscribed in the circle. The center divides this base exactly in half. If the base AB is 12, then the radius R can be found as follows: R=12/2=6.

Answer: The radius is 6.

In geometry, it is important to know the formulas. But it is impossible to remember all of them, so even in many exams it is allowed to use a special form. However, it is important to be able to find correct formula to solve a particular problem. Practice solving different tasks to find the radius and area of ​​a circle in order to be able to correctly substitute formulas and get accurate answers.

Video: Mathematics | Calculation of the areas of a circle and its parts

Instructions

Use Pi to find the radius of famous square 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. From this it follows that the radius can be expressed as Square root from 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 ancient world. In addition to calculating the size of our planet, he made a number of important inventions and discoveries. Invented a simple method to determine prime numbers, now called the “sieve of Erasstophenes”.

He drew a “map of the world”, in which he showed all parts of the world known to the ancient Greeks at that time. The map was considered one of the best for its time. 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

As we know from school curriculum, a circle is usually called a flat geometric figure, which consists of many points equidistant from the center of the figure. Since they are all at the same distance, they form a circle.

Convenient navigation through the article:

Circle area calculator

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 for correct calculation area of ​​a circle. Besides, given value calculated using 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 its 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 obtained data 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), the area of ​​a circle and a ball, and the volume of a ball.

Calculate radius

The problem 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 Internet 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 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 allowing people to calculate the area of ​​a ball quite accurately 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.

In geometry all around is a set of all points on the plane that are removed from one point, called its center, by a distance not greater than a given one, called its radius. In this case, the outer boundary of the circle is circle, and in the case if the length of the radius is zero, circle degenerates to a point.

Determining the area of ​​a circle

If necessary area of ​​a circle can be calculated using the formula:

S πr 2 D 2

r- circle radius

D- circle diameter

S- area of ​​a circle

π - 3.14

This geometric figure is very often found both in technology and in architecture. Designers of machines and mechanisms develop various parts, the sections of many of which are exactly circle. For example, these are shafts, rods, rods, cylinders, axles, pistons, and so on. In the manufacture of these parts, blanks from various materials(metals, wood, plastics), their sections also represent exactly circle. It goes without saying that developers often have to calculate area of ​​a circle through diameter or radius, using simple mathematical formulas, discovered in ancient times.

Exactly then round elements began to be actively and widely used in architecture. One of the most striking examples of this is the circus, which is a type of building designed to host various entertainment events. Their arenas are shaped circle, and they first began to be built in ancient times. The word itself " circus"translated from Latin language means " circle" If in ancient times circuses hosted theatrical performances and gladiator fights, now they serve as places where circus performances with the participation of trainers, acrobats, magicians, clowns, etc. are almost exclusively held. Standard diameter The length of the circus arena is 13 meters, and this is not at all accidental: the fact is that it provides the minimum necessary geometric parameters of the arena in which circus horses can gallop in a circle. If we calculate area of ​​a circle through the diameter, it turns out that for a circus arena this value is 113.04 square meters.

Architectural elements that can take the shape of a circle are windows. Of course, in most cases they are rectangular or square (largely due to the fact that this is easier for both architects and builders), but in some buildings you can also find round windows. Moreover, in vehicles such as air, sea and river vessels they are most often like this.

It is by no means uncommon to use round elements for the production of furniture, such as tables and chairs. There is even a concept " round table ", which implies a constructive discussion, during which there is a comprehensive discussion of various important issues and ways to solve them are developed. As for the manufacture of the countertops themselves, which have round shape, then specialized tools and equipment are used for their production, subject to the participation of workers with fairly high qualifications.



Share with friends:
Related publications