Instruction
1
In problems of plane geometry is necessary to find the area of a polygon inscribed in a circle, or described about him. A polygon is considered to be described about the circle, if he is outside, and its sides touch the circle. The polygon inside the circle is considered to be inscribed in it, if its vertices lie on the circumference of the circle. If in the problem given a triangle inscribed in a circle that all three vertices touch the circle. Depending on what is considered a triangle, and selects the method of solving the problem.
2
The simplest case occurs when the circumference of the inscribed regular triangle. Since such a triangle all sides are equal, the radius of the circle is equal to half its height. Therefore, knowing the sides of a triangle, find its area. To calculate this area, in this case, you can use any of ways, for example:
R=abc/4S, where S is the area of the triangle a, b, c be the sidelengths of triangle
S=0.25(R/abc)
R=abc/4S, where S is the area of the triangle a, b, c be the sidelengths of triangle
S=0.25(R/abc)
3
Another situation occurs when the triangle is isosceles. If the base of the triangle coincides with the diameter line of the circle or diameter is the height of the triangle, the area can be calculated in the following way:
S=1/2h*AC, where AC is the base of the triangle
If you know the radius of the circumcircle of an isosceles triangle, its angles, and the base coinciding with the diameter of a circle, the Pythagorean theorem can be found unknown height. The area of a triangle, whose base coincides with the diameter of a circle is equal to:
S=R*h
In another case, when the height is equal to the diameter of a circle circumscribed around an isosceles triangle, its area is equal to:
S=R*AC
S=1/2h*AC, where AC is the base of the triangle
If you know the radius of the circumcircle of an isosceles triangle, its angles, and the base coinciding with the diameter of a circle, the Pythagorean theorem can be found unknown height. The area of a triangle, whose base coincides with the diameter of a circle is equal to:
S=R*h
In another case, when the height is equal to the diameter of a circle circumscribed around an isosceles triangle, its area is equal to:
S=R*AC
4
The number of tasks in the circle is inscribed right triangle. In this case, the center of the circle lies in the middle of the hypotenuse. Knowing the angles and finding the base of the triangle, you can calculate the area of any of the methods described above.
In other cases, especially when the triangle is acute-angled or obtuse-angled, applicable only the first of the above formulas.
In other cases, especially when the triangle is acute-angled or obtuse-angled, applicable only the first of the above formulas.