How to find the radius of a circle? This question is always relevant for students studying planimetry. Below we will look at several examples of how to cope with the task.
Depending on the condition of the problem, the radius of the circle can be found as follows.
Formula 1: R = / 2Ο, where is the circumference, and Ο is a constant equal to 3.141 ...
Formula 2: R = β (S / Ο), where S is the size of the circle area.
Formula 3: R = D / 2, where D is the diameter of the circle, that is, the length of the segment that, passing through the center of the figure, connects two points as far from each other as possible.
How to find the radius of the circumscribed circle
First, let's define the term itself. A circle is called described when it touches all the vertices of a given polygon. It should be noted that a circle can be described only around such a polygon, the sides and angles of which are equal to each other, that is, around an equilateral triangle, square, regular rhombus, etc. To solve this problem, it is necessary to find the perimeter of the polygon, as well as measure its sides and area. Therefore, arm yourself with a ruler, a compass, a calculator and a notebook with a pen.
How to find the radius of a circle if it is described around a triangle
Formula 1: R = (A * B * C) / 4S, where A, B, C are the lengths of the sides of the triangle, and S is its area.
Formula 2: R = A / sin a, where A is the length of one of the sides of the figure, and sin a is the calculated value of the sine of the angle opposite to this side.
The radius of the circle that is described around a right triangle.
Formula 1: R = B / 2, where B is the hypotenuse.
Formula 2: R = M * B, where B is the hypotenuse, and M is the median drawn to it.
How to find the radius of a circle if it is described around a regular polygon
Formula: R = A / (2 * sin (360 / (2 * n))), where A is the length of one of the sides of the figure, and n is the number of sides in the given geometric figure.
How to find the radius of an inscribed circle
An inscribed circle is called when it touches all sides of the polygon. Let's look at a few examples.
Formula 1: R = S / (P / 2), where - S and P are the area and perimeter of the figure, respectively.
Formula 2: R = (P / 2 - A) * tg (a / 2), where P is the perimeter, A is the length of one of the sides, and is the angle opposite to this side.
How to find the radius of a circle if it is inscribed in a right triangle
Formula 1:
Radius of a circle inscribed in a rhombus
The circle can be entered in any rhombus, both equilateral and non-equilateral.
Formula 1: R = 2 * H, where H is the height of the geometric figure.
Formula 2: R = S / (A * 2), where S is the area of ββthe rhombus, and A is the length of its side.
Formula 3: R = β ((S * sin A) / 4), where S is the area of ββthe rhombus, and sin A is the sine of the acute angle of the given geometric figure.
Formula 4: R = * / (β (Β² + Β²), where and are the lengths of the diagonals of the geometric figure.
Formula 5: R = B * sin (A / 2), where B is the diagonal of the rhombus, and A is the angle at the vertices connecting the diagonal.
Radius of a circle inscribed in a triangle
In the event that, in the condition of the problem, you are given the lengths of all sides of the figure, then first calculate the perimeter of the triangle (P), and then the half-perimeter (p):
P = A + B + C, where A, B, C are the lengths of the sides of the geometric figure.
n = n / 2.
Formula 1: R = β ((p-A) * (p-B) * (p-B) / p).
And if, knowing all the same three sides, you are also given the area of ββthe figure, then you can calculate the desired radius as follows.
Formula 2: R = S * 2 (A + B + C)
Formula 3: R = S / n = S / (A + B + C) / 2), where - n is the semi-perimeter of the geometric figure.
Formula 4: R = (n - A) * tg (A / 2), where n is the semiperimeter of the triangle, A is one of its sides, and tg (A / 2) is the tangent of half of the angle opposite this side.
And the formula below will help you find the radius of the circle that is inscribed in an equilateral triangle.
Formula 5: R = A * β3 / 6.
Radius of a circle inscribed in a right triangle
If the lengths of the legs, as well as the hypotenuse are given in the problem, then the radius of the inscribed circle is recognized as follows.
Formula 1: R = (A + B-C) ββ/ 2, where A, B - legs, C - hypotenuse.
In the event that you are given only two legs, it is time to recall the Pythagorean theorem in order to find and use the above formula.
C = β (AΒ² + BΒ²).
Radius of a circle inscribed in a square
The circle, which is inscribed in the square, divides all its 4 sides exactly in half at the points of tangency.
Formula 1: R = A / 2, where A is the length of the side of the square.
Formula 2: R = S / (P / 2), where S and P are the area and perimeter of the square, respectively.