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

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 the diameter, which 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 above formula: 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.

Knowing the standard formulas described above for the correct calculation of 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 equation into the formula for calculating the area of ​​a circle, and as a result we get the 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 in 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.

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 with finding the 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 described 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 bounded 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 times 2.

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

The formula for finding the area of ​​a circle using the radius:

Examples of solving tasks for finding the S-area of ​​a circle through the R-radius:

A 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 in terms of the D-diameter is:

Examples of solving tasks for finding S, if D is known:

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A task: Find the S of the 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 round figure is 78.5 cm².

Finding the S circle if the circumference is known:

First, find what the radius is. The circumference 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.

Consider the solution on the example of the problem:

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A 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 ​​a circle is 11.46 cm².

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

The formula for finding the area of ​​a circle inscribed in a square is:

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

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Task #1: The side of a square figure is known, which is equal to 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 round figure is 28.26 cm².

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Task #2: Find S of a circle inscribed in a square figure and its radius if one side is a=4 cm.

Decide like this: First 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 round figure is 12.56 cm².

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

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

Examples of solving tasks for finding the area of ​​a circle described near a square figure:

A task

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

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

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

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

Examples of solving tasks:

Task #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 #2

Solution:

Now that you know the radius, you can find the area of ​​the circle in terms of the radius. See the formula above.

Task #3

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

All formulas for finding the area of ​​a circle come 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-angled 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 the 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. Consider 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 the lateral side is known, then you can find the radius through this value. The height of the side of the 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, only an isosceles trapezoid can be inscribed. 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 the circle is inscribed isosceles trapezoid. 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 the right formula for solving 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 | Calculating the area of ​​a circle and its parts

• The length of the diameter - a segment passing through the center of the circle and connecting two opposite points of the circle, or the radius - a segment, one of extreme points which is located in the center of the circle, and the second - on the arc of the circle. So the diameter equal to length radius multiplied by two.
• The value of the number π. This value is a constant - an irrational fraction that has no end. However, it is not periodic. This number expresses the ratio circumference to its radius. To calculate the area of ​​a circle in the tasks of the school course, the value of π is used, given to the nearest hundredth - 3.14.

Formulas for finding the area of ​​a circle, its segment or sector

Depending on the specifics of the conditions of the geometric problem, two formulas for finding the area of ​​a circle:

To determine how to find the area of ​​a circle in the easiest way, you need to carefully analyze the conditions of the task.

The school geometry course also includes tasks for calculating the area of ​​\u200b\u200bsegments or sectors for which special formulas are used:

1. A sector is a part of a circle bounded by a circle and an angle with the vertex located in the center. The area of ​​the sector is calculated by the formula: S = (π*r 2 /360)*А;
   • r is the radius;
   • A is the angle in degrees.
   • r is the radius;
   • p is the length of the arc.

There is also a second option S = 0.5 * p * r;

2. Segment - is a part bounded by a section of a circle (chord) and a circle. Its area can be found by the formula S \u003d (π * r 2 / 360) * A ± S ∆ ;

• r is the radius;
• A is the angle value in degrees;
• S ∆ is the area of ​​a triangle, the sides of which are the radii and the chord of the circle; moreover, one of its vertices is located in the center of the circle, and the other two are located at the points of contact of the arc of the circle with the chord. Important point- the minus sign is placed if the value of A is less than 180 degrees, and the plus sign - if it is more than 180 degrees.

To simplify the solution of a geometric problem, one can calculate circle area online. A special program will quickly and accurately make the calculation in a couple of seconds. How to calculate the area of ​​figures online? To do this, you need to enter the known initial data: radius, diameter, angle.

Instruction

Use pi to find the radius famous area circle. This constant specifies the proportion between the diameter of a circle and the length of its border (circle). Circumference maximum area plane, which it is possible to cover with its help, and the diameter is equal to two radii, therefore, the area with the radius also correlate with each other with a proportion that can be expressed in terms of 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 Square root from the quotient of dividing the area by Pi: r=√(S/π).

For a long time, Erastofen headed the Library of Alexandria, the most famous library ancient world. In addition to the fact that he calculated the size of our planet, he made another series important inventions and discoveries. Invented a simple method to determine prime numbers, now called "Erastothenes' sieve".

He drew a "map of the world", in which he showed all parts of the world known at that time to the ancient Greeks. 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 mechanical device used by early astronomers to demonstrate and predict the apparent movement of stars in the sky. He also compiled a star catalog, which included 675 stars.

Sources:

• The Greek scientist Eratosthenes of Cyrene for the first time in the world calculated the radius of the Earth
• Eratosthenes "Calculation of Earth" s Circumference
• Eratosthenes

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

Determining the area of ​​a circle

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

r- circle radius

D- circle diameter

S- area of ​​a circle

π - 3.14

This geometric figure very common in both engineering and architecture. Designers of machines and mechanisms develop various parts, the sections of many of which are precisely a circle. For example, these are shafts, rods, rods, cylinders, axles, pistons, and so on. In the manufacture of these parts, blanks are used various materials(metals, wood, plastics), their sections also represent precisely a circle. It goes without saying that developers often have to calculate area of ​​a circle through a 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 kind of buildings designed to host various entertainment events. Their arenas are shaped circle, and for the first time they began to be built in antiquity. The very word " circle» translated from Latin means " a circle". If in ancient times circuses hosted theatrical performances and gladiator fights, now they serve as a place where circus performances are almost exclusively held with the participation of animal trainers, acrobats, magicians, clowns, etc. The standard diameter of the circus arena is 13 meters, and this is completely not by chance: the fact is that it is he who provides the minimum necessary geometric parameters an arena where circus horses can gallop in circles. If we calculate area of ​​a circle through the diameter, it turns out that for the circus arena this value is 113.04 square meters.

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

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 a comprehensive discussion of various important problems takes place and ways to solve them are developed. As for the manufacture of the countertops themselves, which have a round shape, they are used for their production specialized tools and equipment, subject to the participation of fairly highly skilled workers.