How to find the area of section

How to find the area of section

Set of tasks in geometry are based on determination of the area of section of a solid. One of the most found solids is the sphere, and determination of the area of its section can prepare for the solution of problems of the most different levels of complexity.

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Instruction

1
Before solving a problem of finding of the area of section, precisely present a required solid, and also constructions, additional to it. For this purpose make the evident drawing of a sphere and construct a secant the area.

2
Put down the conditional parameters designating the radius of a sphere (R), distance between a secant the plane and the center of a sphere (k), the radius of a secant of the area (r) and the required area of the section (S) on the drawing.

3
Define borders of an arrangement of the area of section as the value which is ranging from 0 to? R^2. This interval is caused by two logical conclusions.

- If the distance of k equals to the radius of a secant of the plane, so the plane can concern a sphere only in one point and S equals 0.
- If the distance of k equals 0, then the center of the plane coincides with the center of a sphere, and plane radius – with R radius. Then S find on a formula for calculation of the area of a circle? R^2.

4
Accepting as the fact that a sphere section figure is always the circle, reduce a task to finding of the area of this circle, to be exact to finding of radius of a circle of section. For this purpose present that all points on a circle are tops of a rectangular triangle. As a result of R is a hypotenuse, r – one of legs. K distance – a perpendicular piece which connects a section circle to the center of a sphere becomes the second leg.

5
Considering that other parties of a triangle – a leg of k and a hypotenuse of R – are already set, use Pythagorean theorem. Length of a leg of r equals to a square root from expression (R^2 - k^2).

6
Substitute the found value r in a formula for calculation of the area of a circle? R^2. Thus, the area of section S is determined by a formula? (R^2 - k^2). This formula will be true and for boundary points of an arrangement of the area, when k = R or k = 0. At substitution of these values the area of section S equals either 0, or the areas of a circle with R sphere radius.