How to calculate the segment area and area of ​​the sphere segment

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The mathematical value of the area is known fromtimes of ancient Greece. Even in those distant times, the Greeks found out that the area is a continuous part of the surface, which is bounded from all sides by a closed contour. This is a numerical value that is measured in square units. Area is a numerical characteristic of both planar geometric figures (planimetric) and surfaces of bodies in space (volume).

Currently, it is not only found inwithin the framework of the school curriculum in the lessons of geometry and mathematics, but also in astronomy, everyday life, in construction, in engineering development, in production and in many other spheres of human activity. Very often to calculate the areas of segments, we resort to the backyard when decorating the landscape area or when repairing the ultramodern design of the room. Therefore, the knowledge of methods for calculating the area of ​​various geometric figures will be useful always and everywhere.

To calculate the area of ​​a circular segment and a sphere segment, it is necessary to understand the geometric terms that will be needed in the computational process.

First of all, a segment of a circle is a fragmenta flat circle figure that is located between the arc of the circle and the chord that cuts it. Do not confuse this concept with the figure of the sector. These are completely different things.

A chord is a segment that connects two points lying on a circle.

The central angle is formed between two segments - radii. It is measured in degrees by the arc, on which it rests.

The sphere segment is formed by cutting off somethe plane of the part of the sphere (sphere). In this case, the base of the spherical segment is a circle, and the height is a perpendicular that extends from the center of the circle to the intersection with the surface of the sphere. This intersection point is called the vertex of the segment of the sphere.

In order to determine the area of ​​a segmentsphere, you need to know the circumference of the cut circle and the height of the ball segment. The product of these two components will be the area of ​​the sphere segment: S = 2πRh, where h is the segment height, 2πR is the circumference, and R is the radius of the large circle.

In order to calculate the area of ​​a segment of a circle, one can resort to the following formulas:

1.To find the segment area in the simplest way, it is necessary to calculate the difference between the area of ​​the sector into which the segment is inscribed and the area of ​​the isosceles triangle whose base is the chord of the segment: S1 = S2-S3, where S1 is the segment area, S2 is the sector area and S3 - area of ​​a triangle.

One can use the approximate formulacalculating the area of ​​the circular segment: S = 2/3 * (a * h), where a is the base of the triangle or the length of the chord, h is the height of the segment, which is the result of the difference between the radius of the circle and the height of the isosceles triangle.

2. The area of ​​the segment that differs from the semicircle is calculated as follows: S = (π R2: 360) * α ± S3, where π R2 is the area of ​​the circle, α is the degree measurethe central angle that contains the arc of the segment of the circle, S3 is the area of ​​the triangle that is formed between the two radii of the circle and the chord that has an angle at the central point of the circle and two vertices at the points of contact between the radii and the circle.

If the angle α <180 degrees, the minus sign is used, if α> 180 degrees, the plus sign is used.

3.Calculate the area of ​​the segment can be and other methods using trigonometry. As a rule, the triangle is taken as a basis. If the central angle is measured in degrees, then the following formula is acceptable: S = R2 * (π * (α / 180) - sin α) / 2, where R2 is the square of the radius of the circle, α is the degree measure of the central angle.

4.To calculate the area of ​​a segment using trigonometric functions, another formula can be used, provided that the central angle is measured in radians: S = R2 * (α-sin α) / 2, where R2 is the square of the radius of the circle, α is the degree measure of the central angle .

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