Question

Дан ромб известна сторона и острый угол в градусах. Найти R вписанной окружности. Написать формулу нахождения радиуса

Answer 1

Ромб АВСД. Пересечение биссектрис (диагоналей ромба)- точка О.
Угол АВС=β , АВ=ВС=СД=АД.
ΔАОВ - прямоугольный. Его высота h, проведённая из точки О - есть
радиус r вписанной окружности. 
<ABO=β/2, AB=a .--->  OB=a*cosβ/2 ; OA=a*sinβ/2 .
S(ΔAOB)=1/2* OA*OB=a²/2*sinβ/2*cosβ/2=a²/4*sinβ
S(ΔAOB)=1/2*AB*r*=1/2*r*a

$\frac{1}{2}ra=\frac{a^2}{4}\cdot sin \beta \\\\r=\frac{2a^2sin \beta }{4a}=\frac{a\cdot sin \beta }{2}$

Answer 2

Finding the Radius of an Inscribed Circle in a Rhombus

To find the radius of the inscribed circle in a rhombus, we need to know the length of one side of the rhombus and the measure of one acute angle in degrees.

Let's denote the side length of the rhombus as s and the measure of the acute angle as θ.

The formula for finding the radius R of the inscribed circle in a rhombus is:

R = (s/2) * tan(θ/2)

Where: - s is the length of one side of the rhombus. - θ is the measure of one acute angle in degrees. - tan is the tangent function.

Please note that the angle θ should be in radians for the formula to work. If the angle is given in degrees, we need to convert it to radians by multiplying it by π/180.

Let's use this formula to find the radius of the inscribed circle in the given rhombus.

Example Calculation

Let's say the side length of the rhombus is 10 units and the measure of the acute angle is 60 degrees.

Using the formula, we can calculate the radius as follows:

R = (10/2) * tan((60 * π/180)/2)

Simplifying the equation:

R = 5 * tan(π/6)

Using a calculator, we can find that tan(π/6) ≈ 0.5774.

Substituting the value:

R ≈ 5 * 0.5774 ≈ 2.887 units

Therefore, the radius of the inscribed circle in the given rhombus is approximately 2.887 units.

Please note that the actual values of the side length and angle may differ in your specific case. Make sure to substitute the correct values into the formula for an accurate result.

I hope this helps! Let me know if you have any further questions.