Advice 1: How to find the radius described about the triangle circle

For each triangle there is only one circumscribed circle. It's such a circle on which lie all the three vertices of the triangle with the given parameters. Find its radius may be needed not only on the geometry lesson. This has continually encounter designers, cutters, fitters and representatives of many other professions. In order to find its radius, you must know the parameters of the triangle and its properties. The circumcenter is the intersection of all three that we had triangle.
The ability to find the radius of the circle required the designer
You will need
  • The triangle with the given parameters
  • The compass
  • Line
  • Gon
  • Table of sines and cosines
  • Mathematical concepts
  • Determination of the height of the triangle
  • Formulas of sines and cosines
  • Formula of area of triangle
Draw a triangle with the desired settings. The triangle can be build either on three sides or by two sides and angle between them, or a side and two adjacent angles thereto. Label the vertices of the triangle as A, b and C, the angles as α, β, and γ, and opposite the vertex angle side as a, b and c.
Draw a triangle with the given parameters
Guide height to all sides of the triangle and find the point of intersection. Label the height as h with the corresponding sides of indices. Find the point of intersection and label it O. She will be the center of the circle. Thus, the radius of this circle will be the segments OA, Ob and OC.
The radius of the circumscribed circle can be found in two formulas. For one you must first calculate the area of a triangle. It is equal to the product of all the sides of a triangle the sine of any of the angles divided by 2.


In this case, the radius of the circumscribed circle is calculated by the formula


For other formulas, it is sufficient to know the length of one side and the sine of its opposite angle.


Calculate the radius and describe a triangle around the circumference.
Useful advice
Remember, what is the height of the triangle. It is the perpendicular drawn from angle to opposite side.

The area of a triangle can be represented as the product of the square of one of the parties to the sines of the two adjacent angles, divided by the doubled sine of the sum of these angles.

Advice 2: How to find the circumcenter

Sometimes around a convex polygon you can draw a circle so that the tops of all the angles lying on it. Such a circle relative to the polygon should be called described. Her center does not have to be inside the perimeter of the inscribed figure, but using the described properties of the circle, to find this point is usually not very difficult.
How to find the circumcenter
You will need
  • Ruler, pencil, protractor or straight edge, a compass.
If the polygon about which it is necessary to describe a circle drawn on paper, for finding the centreof the circle and quite a ruler, pencil and protractor or angle. Measure the length of any of sides of a shape, determine its middle and put in this part of the drawing auxiliary point. With the help of the set square or the protractor guide on the inside of the polygon perpendicular to the side segment to the intersection with the opposite side.
Do the same thing with any side of the polygon. The intersection of the two constructed segments is the required point. This follows from basic properties are described of a circle - its center in a convex polygon with any number of sides, always lies at the intersection of middle perpendiculars drawn to these parties.
For the regular polygons defining the center ofa inscribed circle can be much easier. For example, if it is a square, then draw two diagonals and their intersection will be the centerom of the inscribed circle. The regular polygon with any even number of sides, it is sufficient to connect the auxiliary segments two pairs lying opposite each other at the corners - center of the described circle must coincide with the point of their intersection. In a right triangle to solve the problem just define the middle of the longest side of the figure is the hypotenuse.
If conditions are unknown, whether it is possible in principle to draw a circumscribed circle for that polygon, after determining the proposed point centerand any of the described ways you can find out. Mark on the compass the distance between the found point and any vertices, set the compass at the intended center of the circle and draw a circle, every vertex must lie on this circle. If not, then not running one of the basic properties and describe a circle about a given polygon is impossible.
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