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. 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 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 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 through the actual circumference by substituting the expression through 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.

Circles require a more careful approach and are much less common in tasks B5. At the same time, general scheme the solutions are 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 consider specific examples radius calculations:

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 absolutely new type tasks, there was nothing like this 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 smaller partition in any Unified State Exam task no in math! 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 required 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 “scrap” sectors (see figure). Therefore, the area of ​​the required sector is 3 6 = 18.

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:

r- circle radius

D- circle diameter

S- area of ​​a circle

π - 3.14

This geometric figure 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) are used; 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 for this purpose simple mathematical formulas discovered in ancient times.

That's when 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 a place where circus performances with the participation of trainers, acrobats, magicians, clowns, etc. are almost exclusively held. The standard diameter of a 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 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 such vehicles, like air, sea and river vessels, they are most often exactly 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.

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 circles. 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:

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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 look at the solution using an example problem:

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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:

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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.

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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 to find 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 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 a rectangular trapezoid is done using the following formula:

If the lateral side is known, then the radius can be found using 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 enter 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 various problems to find the radius and area of ​​a circle so that you can correctly substitute formulas and get accurate answers.

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

• The length of the diameter - a segment passing through the center of a circle and connecting two opposite points of the circle, or the radius - a segment, one of extreme points which is in the center of the circle, and the second is on the arc of the circle. Thus, the diameter is equal to the length of the 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 tasks school course the value of π is used, given with an accuracy of hundredths - 3.14.

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

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

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

The school geometry course also includes tasks on calculating the area of ​​segments 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 sector area is calculated using the formula: S = (π*r 2 /360)*A;
   • r – radius;
   • A is the magnitude of the angle in degrees.
   • r – radius;
   • p – arc length.

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

2. Segment – ​​is a part limited by a section of a circle (chord) and a circle. Its area can be found using the formula S=(π*r 2 /360)*A ± S ∆ ;

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

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