How to find the radius of a circle: to help students
How to find the radius of a circle? This question is always relevant for schoolchildren who study planimetry. Below we will consider several examples of how to cope with the task.
Depending on the condition of the problem, you can find the radius of the circle like this.
Formula 1: R = A / 2π, where A 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 the 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 figure, connects two points that are as far apart as possible from each other.
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 vertices of a given polygon. It should be noted that it is possible to describe a circle only around such a polygon, the sides and angles of which are equal to each other, that is, around an equilateral triangle, a square, a regular rhombus, and so on. To solve the problem, it is necessary to find the perimeter of the polygon, and also to 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 * B) / 4S, where A, B, B - the length of the sides of the triangle, and S - its area.
Formula 2: R = A / sin a, where A is the length of one side 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, 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 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 side of the figure, and n is the number of sides in a given geometric figure.
How to find the radius of an inscribed circle
An inscribed circle is called when it touches all sides of a polygon. Let's consider some 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 - perimeter, A - the length of one side, and - 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, which is inscribed in the rhombus
The circle can be inscribed 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 the diamond and A is the length of its side.
Formula 3: R = √ ((S * sin A) / 4), where S is the diamond area, and sin A is the sine of the acute angle of the given geometric figure.
Formula 4: R = B * Г / (√ (В² + Г²), where В and Г 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.
The radius of the circle that is inscribed in the triangle
If 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 semiperimeter (n):
P = A + B + B, where A, B, B are the lengths of the sides of the geometric figure.
n = n / 2.
Formula 1: R = √ ((n-A) * (n-B) * (n-B) / n).
And if, knowing all the same three sides, you are given the area of the figure, then you can calculate the desired radius as follows.
Formula 2: R = S * 2 (A + B + B)
Formula 3: R = S / n = S / (A + B + B) / 2), where - n - is the semiperimeter of the geometric figure.
Formula 4: R = (n - A) * tg (A / 2), where n is the half -perimeter of the triangle, A is one of its sides, and tg (A / 2) is the tangent of the half of the angle opposite to 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 * √3 / 6.
The radius of the circle, which is inscribed in a right triangle
If in the problem the lengths of the legs are given, as well as the hypotenuse, then the radius of the inscribed circle is recognized as follows.
Formula 1: R = (A + B-C) / 2, where A, B - the legs, C - the hypotenuse.
In the event that you are given only two legs, it's time to remember the Pythagorean theorem so that the hypotenuse can find and use the above formula.
C = √ (A² + B²).
The radius of the circle, which is inscribed in the 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 - 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.
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