Formula for finding the area of a circle. Circle area: formula
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). The circumference of a circle is the maximum area of the plane that can be covered with it, 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 a number of 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, a 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
Circles require a more careful approach and are much less common in B5 tasks. At the same time, the general solution scheme is even simpler than in the case of polygons (see the lesson “Polygon areas 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 it suffices to find R 2 for the solution.
To find the indicated values, it is enough to indicate on the circle a point lying at the intersection of the grid lines. And then use the Pythagorean theorem. Consider concrete examples radius calculations:
A 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 completes the figure up to right triangle. It remains to find the radii:
Consider triangle ABC in the first circle. According to the Pythagorean theorem: R 2 \u003d AB 2 \u003d AC 2 + BC 2 \u003d 2 2 + 2 2 \u003d 8.
For the second circle, everything is obvious: R = AB = 2.
The third case is similar to the first. From triangle ABC according to the Pythagorean theorem: R 2 \u003d AB 2 \u003d AC 2 + BC 2 \u003d 1 2 + 2 2 \u003d 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 tasks where it is required 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 is this sector, and thus find the area.
A task. Find the area S of the shaded sector. Indicate S / π in your answer.
Obviously, the sector is one quarter of the circle. Therefore, S = 0.25 S of the circle.
It remains to find the S of the circle - the area of the circle. To do this, we will perform an additional construction:
Triangle ABC is a right triangle. By the Pythagorean theorem, we have: R 2 \u003d AB 2 \u003d AC 2 + BC 2 \u003d 2 2 + 2 2 \u003d 8.
Now we find the area of the circle and the sector: S of the circle = πR 2 = 8π; S = 0.25 S circle = 2π.
Finally, the desired value is equal to S /π = 2.
Sector area with unknown radius
This is a completely new type of task, there was nothing like it in 2010-2011. By condition, we are given a circle of a certain area (namely, the area, not the radius!). Then, inside this circle, a sector is allocated, the area of which is required to be found.
The good news is that these problems are the easiest of all the problems in the square, which are in the exam in mathematics. In addition, the circle and sector are always placed on the coordinate grid. Therefore, to learn how to solve such problems, just take a look at the picture:
Let the original circle have area S of the circle = 80. Then it can be divided into two sectors of area S = 40 each (see step 2). Similarly, each of these "half" sectors can be divided in half again - we get four sectors of area S = 20 each (see step 3). Finally, you can divide each of these sectors into two more - we get 8 sectors - "little pieces". The area of each of these "chunks" will be S = 10.
Please note: a finer partition in none USE task no math! Thus, the algorithm for solving problem B-3 is as follows:
- Cut the original circle into 8 sectors - "pieces". 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 the area S of the circle = 240, then the “lumps” have the area S = 240: 8 = 30;
- Find out how many "lumps" fit in the original sector, the area of which you want to find. For example, if our sector contains 3 “lumps” with an area of 30, then the area of the desired sector is S = 3 30 = 90. This will be the answer.
That's all! The problem is solved practically orally. If you still don't understand something, buy a pizza and cut it into 8 pieces. Each such piece will be the same sector - "chunk" that can be combined into larger pieces.
And now let's look at examples from the trial exam:
A task. A circle with an area of 40 is drawn on checkered paper. Find the area of the shaded figure.
So, the area of the circle is 40. Divide it into 8 sectors - each with an area of S = 40: 5 = 8. We get:
Obviously, the shaded sector consists of exactly two "small" sectors. Therefore, its area is 2 5 = 10. That's the whole solution!
A task. A circle with an area of 64 is drawn on checkered paper. Find the area of the shaded figure.
Again, divide the entire circle into 8 equal sectors. Obviously, the area of one of them just needs to be found. Therefore, its area is S = 64: 8 = 8.
A task. A circle with an area of 48 is drawn on checkered paper. 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. Exactly three sectors-“small” are placed in the desired sector (see figure). Therefore, the area of the desired sector is 3 6 = 18.
- this is flat figure, which is a set of points equidistant from the center. All of them are at the same distance and form a circle.
A line segment that connects the center of a circle with points on its circumference is called radius. In each circle, all radii are equal to each other. A line joining two points on a circle and passing through the center is called diameter. The formula for the area of a circle is calculated using a mathematical constant - the number π ..
It is interesting : The number pi. is the ratio of the circumference of a circle to the length of its diameter and is a constant value. The value π = 3.1415926 was used after the work of L. Euler in 1737.
The area of a circle can be calculated using the constant π. and the radius of the circle. The formula for the area of a circle in terms of radius looks like this:
Consider an example of calculating the area of a circle using the radius. Let a circle with radius R = 4 cm be given. Let's find the area of the figure.
The area of our circle will be equal to 50.24 square meters. cm.
There is a formula the area of a circle through the diameter. It is also widely used to calculate the required parameters. These formulas can be used to find .
Consider an example of calculating the area of a circle through the diameter, knowing its radius. Let a circle be given with a radius R = 4 cm. First, let's find the diameter, which, as you know, is twice the radius.
Now we use the data for the example of calculating the area of a circle using the above formula:
As you can see, as a result we get the same answer as in the first calculations.
Knowledge of the standard formulas for calculating the area of a circle will help in the future to easily determine sector area and it is easy to find the missing quantities.
We already know that the formula for the area of a circle is calculated through the product of the constant value π and the square of the radius of the circle. The radius can be expressed in terms of the circumference of a circle and substitute the expression in the formula for the area of a circle in terms of the circumference:
Now we substitute this equality into the formula for calculating the area of a circle and get the formula for finding the area of \u200b\u200bthe circle, through the circumference
Consider an example of calculating the area of a circle through the circumference. Let a circle be given with length l = 8 cm. Let's substitute the value in the derived formula: 
The total area of the circle will be 5 square meters. cm.
Area of a circle circumscribed around a square
It is very easy to find the area of a circle circumscribed around a square.
This will require only the side of the square and knowledge of simple formulas. The diagonal of the square will be equal to the diagonal of the circumscribed circle. Knowing the side a, it can be found using the Pythagorean theorem: from here.
After we find the diagonal, we can calculate the radius: .
And then we substitute everything into the basic formula for the area of a circle circumscribed around a square:
A circle is a visible collection of many points that are at the same distance from the center. To find its area, you need to know what the radius, diameter, π number and circumference are.
Quantities involved in calculating the area of a circle
The distance bounded by the center point of the circle and any of the points on the circle is called the radius of this circle. geometric figure. The lengths of all radii of one circle are the same. The line segment between any 2 points on the circle that passes through the center point is called the diameter. The length of the diameter is equal to the length of the radius multiplied by 2.
To calculate the area of a circle, the value of the number π is used. This value is equal to the ratio of the circumference to the length of the diameter of the circle and has a constant value. Π = 3.1415926. The circumference is calculated using the formula L=2πR.
Find the area of a circle using the radius
Therefore, the area of a circle is equal to the product of the number π and the radius of the circle raised to the 2nd power. As an example, let's take the length of the radius of the circle equal to 5 cm. Then the area of the circle S will be equal to 3.14 * 5 ^ 2 = 78.5 square meters. cm.
Circle area in terms of diameter
The area of a circle can also be calculated by knowing the diameter of the circle. In this case, S = (π/4)*d^2, where d is the diameter of the circle. Let's take the same example where the radius is 5 cm. Then its diameter will be 5*2=10 cm. The area of the circle is S=3.14/4*10^2=78.5 sq.cm. The result, which is equal to the total of the calculations in the first example, confirms the correctness of the calculations in both cases.
Area of a circle in terms of circumference
If the radius of a circle is represented through the circumference, then the formula will look like this: R=(L/2)π. Substitute this expression into the formula for the area of a circle and as a result we get S=(L^2)/4π. Consider an example in which the circumference is 10 cm. Then the area of the circle is S = (10 ^ 2) / 4 * 3.14 = 7.96 square meters. cm.
Area of a circle in terms of the length of a side of an inscribed square
If a square is inscribed in a circle, then the length of the diameter of the circle is equal to the length of the diagonal of the square. Knowing the size of the side of the square, you can easily find the diameter of the circle by the formula: d ^ 2 \u003d 2a ^ 2. In other words, the diameter to the power of 2 is equal to the side of the square to the power of 2 times 2.
Having calculated the value of the length of the diameter of a circle, you can also find out its radius, and then use one of the formulas for determining the area of a circle.
Sector area of a circle
A sector is a part of a circle bounded by 2 radii and an arc between them. To find out its area, you need to measure the angle of the sector. After that, it is necessary to compose a fraction, in the numerator of which there will be the value of the angle of the sector, and in the denominator - 360. To calculate the area of \u200b\u200bthe sector, the value obtained as a result of dividing the fraction must be multiplied by the area of \u200b\u200bthe circle calculated using one of the above formulas.
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.
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