Circle Square: Formula. What is the area of \u200b\u200ba circle described and inscribed in a square, a rectangular and isosceles triangle, a rectangular, isosceles trapezoid?

How to find a circle area? 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 at the same distance from the central point. A circle is a flat figure, so solving problems with finding the area is simple. In this article we will consider how to find the area of \u200b\u200ba circle inscribed in a triangle, a trapezoid, a square, and described near these figures.

Circle area: formula through the radius, diameter, circle length, examples of problem solving

To find the area of \u200b\u200bthis figure, you need to know what radius, diameter and number π are.

Radius r - This is a distance limited by the center of the circle. The lengths of all R-radius of one circle will be equal.

Diameter d - This is a line between two of any points of the circle, which passes through the central point. The length of this segment is equal to the length of the R-radius, multiplied by 2.

Number π - This is an unchanged value that is 3.1415926. In mathematics, this number is usually rounded to 3.14.

Formula for finding the area of \u200b\u200bthe circle through the radius:

Examples of solving tasks on finding the S-plane of the circle through R-radius:

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Task: Find the area of \u200b\u200bthe circle if its radius is 7 cm.

Solution: S \u003d πr², S \u003d 3.14*7², S \u003d 3.14*49 \u003d 153.86 cm².

Answer: The area of \u200b\u200bthe circle is 153.86 cm².

Formula for finding the S-plane of the circle through the D-diameter:

Examples of solving tasks on finding s if D:

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Task: Find S Circle if its D is 10 cm.

Solution: P \u003d π*d²/4, p \u003d 3.14*10²/4 \u003d 3.14*100/4 \u003d 314/4 \u003d 78.5 cm².

Answer: The area of \u200b\u200ba flat round figure is 78.5 cm².

Finding s circle, if the length of the circle is known:

First we find what the radius is equal to. The length of the circumference is calculated by the formula: l \u003d 2πR, respectively, the radius R will be L/2π. Now we find the area of \u200b\u200bthe circle according to the formula through R.

Consider the solution on the example of the problem:

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Task: Find the area of \u200b\u200bthe circle if the length of the circumference is L - 12 cm.

Solution: First, we find the radius: R \u003d l/2π \u003d 12/2*3.14 \u003d 12/6.28 \u003d 1.91.

Now we find the area through the radius: S \u003d πr² \u003d 3.14*1.91² \u003d 3.14*3.65 \u003d 11.46 cm².

Answer: The area of \u200b\u200bthe circle is 11.46 cm².

Circle area inscribed in a square: formula, examples of problem solving

Finding the area of \u200b\u200ba circle inscribed in a square is simple. The side of the square is the diameter of the circle. To find the radius, you need to divide the side by 2.

Formula for finding the area of \u200b\u200ba circle inscribed in a square:

Examples of solving problems on finding the area of \u200b\u200ba circle inscribed in a square:

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

Solution: S \u003d π (A/2) ² \u003d 3.14 (6/2) ² \u003d 3.14*9 \u003d 28.26 cm².

Answer: The area of \u200b\u200ba flat round figure is 28.26 cm².

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Task number 2: Find C of a circle inscribed in a square figure and its radius if one side is equal to a \u003d 4 cm.

Decide so: First find R \u003d A/2 \u003d 4/2 \u003d 2 cm.

Now find the circle area S \u003d 3.14*2² \u003d 3.14*4 \u003d 12.56 cm².

Answer: The area of \u200b\u200ba flat round figure is 12.56 cm².

The area of \u200b\u200bthe circle described near the square: formula, examples of problem solving

It is a little more difficult to find the area of \u200b\u200ba round figure described near the square. But, knowing the formula, you can quickly calculate this value.

Formula for the location of a circle described near the square figure:

Examples of solving tasks on finding a circle area described near a square figure:

Task

The area of \u200b\u200ba circle inscribed in a rectangular and isosceles triangle: formula, examples of solving problems

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

The formula for finding the area of \u200b\u200ba circle inscribed in an isosceles triangle:

When the radius is known, the area can be calculated by the formula: S \u003d πR².

The formula for finding the area of \u200b\u200ba circle inscribed in a rectangular triangle:

Examples of solving tasks:

Task number 1

If in this task you also need to find a circle area with a radius of 4 cm, then this can be done according to the formula: S \u003d πR²

Task number 2

Solution:

Now that the radius is known, you can find the area of \u200b\u200bthe circle through the radius. See the formula above in the text.

Task number 3

The area of \u200b\u200bthe circle described near the rectangular and isosceles triangle: formula, examples of solving problems

All formulas for finding the area of \u200b\u200bthe circle come down to the fact that first you need to find its radius. When the radius is known, then finding the area is simple, as described above.

The area of \u200b\u200bthe circle described near the rectangular and isosceles triangle is in the following formula:

Examples of problem solving:

Here is another example of solving the problem using the heroic formula.

It is difficult to solve such problems, but they can be mastered if you know all the formulas. Schoolchildren solve such tasks in grade 9.

The area of \u200b\u200ba circle inscribed in a rectangular and isosceles trapezoid: formula, examples of solving problems

In an isosceles trapezoid, two sides are equal. In a rectangular trapezoid, one angle is 90º. Consider how to find the area of \u200b\u200ba circle inscribed in a rectangular and isosceles trapezoid on the example of solving problems.

For example, a circle is inscribed in an isosceles trapezoid, which at the touch point divides one side into segments M and N.

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

Finding the area of \u200b\u200ba circle inscribed in a rectangular trapezoid is carried out according to the following formula:

If the side 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 \u003d D/2.

Examples of problem solving:

The area of \u200b\u200bthe circle described near the rectangular and isosceles trapezoid: formula, examples of solving problems

The trapezoid can be entered into a circle when the sum of its opposing angles is 180º. Therefore, only an equal trapezoid can be entered. The radius for calculating the area of \u200b\u200bthe circle described near the rectangular or isosceles trapezoid is calculated by the following formulas:

Examples of problem solving:

Solution: A large base in this case passes through the center, since an isosceled trapezoid is inscribed in the circle. The center shares this foundation exactly in half. If the base AB is 12, then the radius R can be found like this: R \u003d 12/2 \u003d 6.

Answer: The radius is 6.

In geometry, it is important to know the formulas. But all of them cannot be remembered, so even in many exams it is allowed to use a special form. However, it is important to be able to find the correct formula for solving a particular problem. Train in solving different tasks for finding a radius and a circle area in order to be able to correctly substitute formulas and receive accurate answers.

Video: Mathematics | Calculation of the area of \u200b\u200bthe circle and its parts