How to find the radius of a circle? This question is always relevant for schoolchildren studying planimetry. Below we look at a few examples of how to cope with the task.
Depending on the task condition, the radius of the circle you can find.
Formula 1: R = L / 2π, where L is the length of the circle, and π is a constant equal to 3.141 ...
Formula 2: R = √ (S / π), where S is the area of a circle.
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 shape, connects two points as far from each other.
How to find the radius of the circumcircle
First, let's define the term itself.A circle is called described when it touches all vertices of a given polygon. It should be noted that the circle can be described only around such a polygon, whose sides and angles are equal to each other, that is, around an equilateral triangle, a square, a regular rhombus, etc. To solve the 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, compass, 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 sine value opposite to this side of the angle.
The radius of the circle, which 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 applied 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 the inscribed circle
The inscribed circle is called when it touches all sides of the polygon. Consider a few examples.
Formula 1: R = S / (P / 2), where - S and P - 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:
The radius of the circle that is inscribed in a rhombus
The circle can be inscribed in any rhombus, both equilateral and non-equilateral.
Formula 1: R = 2 * Н, where Н is the height of a geometric figure.
Formula 2: R = S / (A * 2), where S is the rhombus area, and A is the length of its side.
Formula 3: R = √ (((S * sin А) / 4), where S is the rhombus area, and sin A is the sine of the acute angle of the given geometric figure.
Formula 4: R = В * Г / (√ (В² + ²²), where В and D are the lengths of the diagonals of a 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.
The radius of the circle that is inscribed in a triangle
In the event that in the problem statement you are given the lengths of all sides of the figure, then first calculate the perimeter of the triangle (P), and then the semi-perimeter (n):
P = A + B + C, where A, B, C are the lengths of the sides of a geometric figure.
n = n / 2.
Formula 1: R = √ ((pA) * (pB) * (pV) / p).
And if, knowing all the same three sides, you are also given the area of the figure, 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 a semi-perimeter of a geometric figure.
Formula 4: R = (n - A) * tg (A / 2), where n is the semi-perimeter of the triangle, A is one of its sides, and tg (A / 2) is the tangent of half opposite to this side of the angle.
And the formula below will help find the radius of the circle that is inscribed in an equilateral triangle.
Formula 5: R = A * √3 / 6.
The radius of a circle that is inscribed in a right triangle
If the problem is given the length of the legs, as well as the hypotenuse, then the radius of the inscribed circle is recognized as follows.
Formula 1: R = (A + BS) / 2, where A, B are the legs, C is the hypotenuse.
In the event that you are given only two legs, it's time to recall the Pythagorean theorem in order to find the hypotenuse and use the above formula.
C = √ (А² + Б²).
The radius of the circle that is inscribed in a square
The circle, which is inscribed in a 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.
