The mathematical value of the area has been known since ancient Greece. Even in those days, the Greeks found that the area is the solid part of the surface, which is bounded on all sides by a closed loop. This is a numerical value that is measured in square units. Area is a numerical characteristic of both flat geometric figures (planimetric) and surfaces of bodies in space (volumetric).
At present, it is found not only in the framework of the school curriculum at the lessons of geometry and mathematics, but also in astronomy, everyday life, in construction, in design developments, in production and in many other areas of human activity . Very often we resort to calculating the area of segments on a personal plot during the design of the landscape area or during repair work of an ultramodern design of the room. Therefore, knowledge of methods for calculating the area of various geometric shapes 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 computing process.
First of all, a segment of a circle is a fragment of a flat figure of a circle, which is located between the arc of a circle and the chord that cuts it off. This concept should not be confused with a sector figure. These are completely different things.
A chord is a line that connects two points lying on a circle.
The central angle is formed between two segments - radii. It is measured in degrees by an arc against which it rests.
A sphere segment is formed when some part of the ball (sphere) is cut off by a plane . In this case, the base of the spherical segment is a circle, and the height is the perpendicular extending from the center of the circle to the intersection with the surface of the sphere. This intersection point is called the top of the ball segment.
In order to determine the area of a segment of a sphere, you need to know the circumference of the cut-off circle and the height of the spherical segment. The product of these two components will be the area of the sphere segment: S = 2πRh, where h is the height of the segment, 2πR is the circumference, and R is the radius of the big circle.
In order to calculate the area of a circle segment, you can resort to the following formulas:
1. To find the area of a segment 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 an isosceles triangle whose base is the chord of the segment: S1 = S2-S3, where S1 is the area of the segment, S2 is the area of the sector and S3 is the area of the triangle.
You can use the approximate formula for calculating the area of a 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 a segment different from a semicircle is calculated as follows: S = (π R2: 360) * α ± S3, where π R2 is the area of the circle, α is the degree measure of the central angle that contains the arc of the segment of the circle, S3 is the area of the triangle , which was formed between the two radii of the circle and the chord, which has an angle at the center point of the circle and two peaks at the points of contact of the radii with the circle.
If the angle α <180 degrees, the minus sign is used, if α> 180 degrees, the plus sign is used.
3. You can calculate the area of a segment by other methods using trigonometry. As a rule, a 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, one can use another formula 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 central angle.