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. The circle is flat figure, so solving problems with 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 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.
Formula for finding the area of a circle inscribed in 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 consider how to find the area of a circle inscribed in a rectangular and isosceles trapezoid using the example of problem solving.
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 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 the right 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
Instructions
If you have the opportunity to use a protractor when constructing, start by choosing an arbitrary point on the circle, which should become one of the vertices of the correct one. Label it, for example, with the letter A.
Draw an auxiliary segment connecting A to the center of the circle. Attach a protractor to this segment so that the zero division coincides with the center of the circle, and place an auxiliary point at the 120° mark. Through this point, draw another auxiliary segment with the beginning in the center of the circle at the intersection with circumference. Mark the intersection point with the letter B - this is the second vertex of the inscribed triangle.
Repeat the previous step, but apply the protractor to the second auxiliary segment, and the point of intersection with circumference designate it with the letter C. You will no longer need a protractor.
If there is no protractor, but there is a compass and , then start by calculating the length of the side triangle. You probably know that it can be expressed in terms of the radius of the circumscribed circle, multiplying it by triples to square root out of three, that is, by approximately 1.732050807568877. Round this to your desired precision and multiply by the radius of the circle.
Set aside the side length found in the fifth step on the compass. triangle and an auxiliary circle with a center at point A. Designate the intersection points of the two circles with the letters B and C - these are the other two vertices of the regular circle inscribed in the circle triangle.
Connect points A and B, B and C, C and A and the construction will be completed.
If a circle touches all three sides of a given triangle and its center is inside the triangle, then it is called inscribed in the triangle.
You will need
- ruler, compass
Instructions
The point of intersection of the arcs along the ruler is connected to the vertex of the divisible angle;
The same is done with any other angle;
Sources:
- http://www.alleng.ru/d/math/math42.htm
Correct triangle- one in which all sides are the same length. Based on this definition, the construction of such a variety triangle but is not a difficult task.
You will need
- Ruler, sheet of lined paper, pencil
Instructions
note
In a regular (equilateral) triangle, all angles are equal to 60 degrees.
An equilateral triangle is also an isosceles triangle. If a triangle is isosceles, this means that 2 of its 3 sides are equal, and the third side is considered the base. Any regular triangle is isosceles, while the converse is not true.
Tip 4: How to find the area of a triangle inscribed in a circle
The area of a triangle can be calculated in several ways, depending on what value is known from the problem conditions. Given the base and height of a triangle, the area can be found by calculating the product of half the base and the height. In the second method, the area is calculated through the circumcircle of the triangle.
Instructions
In planimetry problems, you have to find the area of a polygon inscribed in a circle or circumscribed around it. A polygon is considered circumscribed about a circle if it is outside and its sides touch the circle. A polygon located inside a circle is considered inscribed in it if its circles lie on it. If the problem is given , which is inscribed, all three of its vertices touch the circle. Depending on what kind of triangle is being considered, the method of the task is chosen.
The simplest case is when a regular triangle is inscribed in. Since such a triangle has everything, the radius of the circle is equal to half its height. Therefore, of a triangle, you can find its area. In this case, you can calculate this area in any of the following ways, for example:
R=abc/4S, where S is the area of the triangle, a, b, c are the sides of the triangle
Another situation arises when the triangle is isosceles. If the base of the triangle coincides with the line of the diameter of the circle or the diameter is also the height of the triangle, the area can be calculated as follows:
S=1/2h*AC, where AC is the base of the triangle
If the radius of a circle, its angles, as well as the base coinciding with the diameter of the circle are known, the unknown height can be found using the Pythagorean theorem. The area of a triangle whose base coincides with the diameter of the circle is:
S=R*h
In another case, when the height is equal to the diameter of the circle circumscribed around an isosceles triangle, its area is equal to:
S=R*AC
In a number of problems, a right triangle is inscribed in a circle. In this case, the center of the circle lies at the middle of the hypotenuse. Knowing the angles and base of a triangle, you can calculate the area using any of the methods described above.
In other cases, especially when the triangle is acute or obtuse, only the first of the above formulas is applicable.
The task is to fit into circle polygon can often confuse an adult. Her decision needs to be explained to a schoolchild, so parents go surfing the World Wide Web in search of a solution.
Instructions
Draw circle. Place the compass needle on the side of the circle, but do not change the radius. Draw two arcs crossing circle, turning the compass to the right and left.
Move the compass needle along the circle to the point where the arc intersects it. Turn the compass again and draw two more arcs, crossing the contour of the circle. Repeat this procedure until it intersects with the first point.
Draw circle. Draw the diameter through its center, the line should be horizontal. Construct a perpendicular to through the center of the circle, get a vertical line (CB, for example).
Divide the radius in half. Mark this point on the diameter line (label it A). Build circle with center at point A and radius AC. When it intersects with a horizontal line, you will get another point (D, for example). As a result, the segment CD will be the side of the pentagon that needs to be inscribed.
Lay semicircles, the radius of which is equal to CD, along the contour of the circle. Thus, the original circle will be divided into five equal parts. Connect the dots with a ruler. The problem of inscribing a pentagon into circle also completed.
The following is described by fitting into circle square. Draw a diameter line. Take a protractor. Place it at the point where the diameter intersects the side of the circle. Open the compass to the length of the radius.
Draw two arcs until they intersect with circle yu, turning the compass in one direction or the other. Move the leg of the compass to the opposite point and draw two more arcs with the same solution. Connect the resulting dots.
Square the diameter, divide by two and take the root. As a result, you will get a side of a square that will easily fit into circle. Open the compass to this length. Put his needle on circle and draw an arc intersecting one side of the circle. Move the leg of the compass to the resulting point. Draw the arc again.
Repeat the procedure and draw two more points. Connect all four dots. This is an easier way to fit a square into circle.
Consider the task of fitting into circle. Draw circle. Take a point arbitrarily on the circle - it will be the vertex of the triangle. From this point, maintaining a compass, draw an arc until it intersects with circle Yu. This will be the second peak. Construct a third vertex from it in a similar way. Connect the dots with a ruler. The solution has been found.
Video on the topic
Being one of the integral parts school curriculum, geometric problems for constructing regular polygons are quite trivial. As a rule, construction is carried out by inscribing a polygon into circle, which is drawn first. But what if circle given, but the figure is very complex?
You will need
- - ruler;
- - compass;
- - pencil;
- - paper.
Instructions
Construct a line segment perpendicular to AB and dividing it into two equal parts at the intersection point. Place the needle of the compass at point A. Place the leg with the lead at point B, or at any point on the segment that is closer to B than to A. Draw circle. Without changing the angle of the legs of the compass, set its needle to point B. Draw another circle.The drawn circles will intersect in two. Draw a straight line through them. Designate the point of intersection of this segment with segment AB as C. Designate the points of intersection of this segment with the original circle yu like D and E.
Construct a line segment DE dividing it in half. Carry out actions similar to those described in the previous step in relation to the segment DE. Let the drawn segment intersect DE at point O. This point will be the center of the circle. Also mark the points of intersection of the constructed perpendicular with the original one circle yu like F and G.
Set the opening of the compass legs so that the distance between their ends is the radius of the original circle. To do this, place the needle of the compass at one of points A, B, D, E, F or G. Place the end of the leg with the lead at point O.
Construct a regular hexagon. Place the compass needle at any point on the circle line. Label this point H. In a clockwise direction, make an arcuate notch with a compass so that it intersects the circle line. Label this point I. Move the compass needle to point I. Make a notch on the circle again and label the resulting point J. Similarly, construct points K, L, M. Consistently connect points H, I, J, K, L, M, H in pairs .Received
In modern mechanical engineering, a lot of elements and spare parts are used, which have both external and internal circles in their structure. The most striking examples are bearing housings, engine parts, hub assemblies and much more. In their production, not only high-tech devices are used, but also knowledge from geometry, in particular information about the circles of a triangle. We will get acquainted with this knowledge in more detail below.
In contact with
Which circle is inscribed and which is circumscribed?
First of all, remember that a circle is an infinite set of points at equal distances from the center. If inside a polygon it is possible to construct a circle that has only one common intersection point with each side, then it will be called inscribed. A circumscribed circle (not a circle, these are different concepts) is a geometric locus of points such that the constructed figure with a given polygon common points there will be only the vertices of the polygon. Let's get acquainted with these two concepts using a more clear example (see Figure 1.).
Figure 1. Inscribed and circumscribed circles of a triangle
The image shows two figures of large and small diameters, the centers of which are G and I. Circle greater value is called the described neighborhood Δ ABC, and the small one is called, on the contrary, inscribed in Δ ABC.
In order to describe the surroundings around a triangle, it is required draw a perpendicular line through the middle of each side(i.e. at an angle of 90°) is the point of intersection, it plays a key role. It will be the center of the circumscribed circle. Before finding a circle, its center in a triangle, you need to construct for each angle, and then select the point of intersection of the lines. It, in turn, will be the center of the inscribed neighborhood, and its radius under any conditions will be perpendicular to any of the sides.
To the question: “How many inscribed circles can there be for a polygon with three?” Let us answer right away that a circle can be inscribed in any triangle, and only one. Because there is only one point of intersection of all bisectors and one point of intersection of perpendiculars emanating from the midpoints of the sides.
Property of the circle to which the vertices of a triangle belong
The circumscribed circle, which depends on the lengths of the sides at the base, has its own properties. Let us indicate the properties of the circumscribed circle:
In order to more clearly understand the principle of the circumscribed circle, let’s solve a simple problem. Let us assume that we are given a triangle Δ ABC, the sides of which are 10, 15 and 8.5 cm. The radius of the circumscribed circle around the triangle (FB) is 7.9 cm. Find the degree measure of each angle and through them the area of the triangle.
Figure 2. Finding the radius of a circle using the ratio of sides and sines of angles
Solution: based on the previously stated theorem of sines, we find the value of the sine of each angle separately. By condition, it is known that side AB is 10 cm. Let’s calculate the value of C:
Using the values of the Bradis table, we find out that the degree measure of angle C is 39°. Using the same method, we can find the remaining measures of angles:
How do we know that CAB = 33°, and ABC = 108°. Now, knowing the values of the sines of each of the angles and the radius, let’s find the area by substituting the found values:
Answer: The area of the triangle is 40.31 cm², and the angles are 33°, 108° and 39°, respectively.
Important! When solving problems of this kind, it will be useful to always have Bradis tables or a corresponding application on your smartphone, since the manual process can take a long time. long time. Also, to save more time, it is not necessary to construct all three midpoints of the perpendicular or three bisectors. Any third of them will always intersect at the point of intersection of the first two. And for an orthodox construction, the third is usually completed. Maybe this is wrong when it comes to the algorithm, but on the Unified State Exam or other exams it saves a lot of time.
Calculating the radius of an inscribed circle
All points of a circle are equally distant from its center at the same distance. The length of this segment (from and to) is called the radius. Depending on what kind of environment we have, there are two types - internal and external. Each of them is calculated using its own formula and is directly related to the calculation of parameters such as:
- square;
- degree measure of each angle;
- side lengths and perimeter.
Figure 3. Location of the inscribed circle inside the triangle
You can calculate the length of the distance from the center to the point of contact on either side in the following ways: h through the sides, sides and corners(for an isosceles triangle).
Using a semi-perimeter
A semiperimeter is half the sum of the lengths of all sides. This method is considered the most popular and universal, because no matter what type of triangle is given according to the condition, it is suitable for everyone. The calculation procedure is as follows:
If given "correct"
One of the small advantages of the "ideal" triangle is that inscribed and circumscribed circles have their center at the same point. This is convenient when constructing figures. However, in 80% of cases the answer is “ugly.” What is meant here is that very rarely the radius of the inscribed neighborhood will be whole, rather the opposite. For simplified calculation, use the formula for the radius of the inscribed circle in a triangle:
If the sides are the same length
One of the subtypes of tasks for the state. exams will be to find the radius of the inscribed circle of a triangle, two sides of which are equal to each other and the third is not. In this case, we recommend using this algorithm, which will significantly save time on searching for the diameter of the inscribed region. The radius of an inscribed circle in a triangle with equal “sides” is calculated by the formula:
We will demonstrate a more clear application of these formulas in the following problem. Let us have a triangle (Δ HJI), into which the neighborhood is inscribed at point K. The length of side HJ = 16 cm, JI = 9.5 cm and side HI is 19 cm (Figure 4). Find the radius of the inscribed neighborhood, knowing the sides.
Figure 4. Finding the value of the radius of the inscribed circle
Solution: to find the radius of the inscribed environment, we find the semi-perimeter:
From here, knowing the calculation mechanism, we find out the following value. To do this, you will need the lengths of each side (given according to the condition), as well as half the perimeter, it turns out:
It follows that the required radius is 3.63 cm. According to the condition, all sides are equal, then the required radius will be equal to:
Provided that the polygon is isosceles (for example, i = h = 10 cm, j = 8 cm), the diameter of the inner circle centered at point K will be equal to:
The problem may contain a triangle with an angle of 90°; in this case, there is no need to memorize the formula. The hypotenuse of the triangle will be equal to the diameter. It looks more clearly like this:
Important! If the task is to find the internal radius, we do not recommend performing calculations using the values of the sines and cosines of angles, the table value of which is not precisely known. If it is impossible to find out the length otherwise, do not try to “pull out” the value from under the root. In 40% of problems, the resulting value will be transcendental (i.e. infinite), and the commission may not count the answer (even if it is correct) due to its inaccuracy or irregular shape submissions. Special attention Pay attention to how the formula for the circumradius of a triangle can be modified depending on the proposed data. Such “blanks” allow you to “see” the scenario for solving a problem in advance and choose the most economical solution.
Inner circle radius and area
To calculate the area of a triangle inscribed in a circle, use only radius and side lengths of the polygon:
If the problem statement does not directly give the value of the radius, but only the area, then the indicated area formula is transformed into the following:
Let us consider the effect of the last formula on more specific example. Suppose that we are given a triangle into which the neighborhood is inscribed. The area of the neighborhood is 4π, and the sides are 4, 5 and 6 cm, respectively. Let's calculate the area of a given polygon by calculating the semi-perimeter.
Using the above algorithm, we calculate the area of the triangle through the radius of the inscribed circle:
Due to the fact that a circle can be inscribed in any triangle, the number of variations in finding the area increases significantly. Those. finding the area of a triangle involves mandatory knowledge the length of each side, as well as the radius value.
Triangle inscribed in a circle geometry grade 7
Right triangles inscribed in a circle
Conclusion
From these formulas you can be sure that the complexity of any problem using inscribed and circumscribed circles lies only in additional actions to find the required values. Problems of this type require only a thorough understanding of the essence of the formulas, as well as the rationality of their application. From the practice of solving, we note that in the future the center of the circumscribed circle will appear in further geometry topics, so it should not be started. Otherwise, the solution may be delayed using unnecessary moves and logical conclusions.