How to find the radius of the circle entered in a rhombus

How to find the radius of the circle entered in a rhombus

The parallelogram which all parties have identical length is called a rhombus. This main property defines also equality of the corners lying in opposite tops of such flat geometrical figure. It is possible to enter a circle which radius pays off in several ways in a rhombus.

Instruction

1. If the area (S) of a rhombus and length of its party (a) is known, then for finding of radius (r) of the circle entered in this geometrical figure calculate private from division of the square at the doubled length of the party: r=S / (2*a). For example, if the area is equal to 150 cm², and length of the party - 15 cm, then the radius of an inscribed circle will be equal 150 / (2*15) = 5 cm.

2. If except the area (S) of a rhombus the size of an acute angle (α) in one of its tops is known, then for calculation of radius of an inscribed circle find a square root from a quarter of the work of the square at a sine of the known corner: r= √ (S*sin (α)/4). For example, if the area is equal to 150 cm², and the known corner has size 25 °, then calculation of radius of an inscribed circle will look so: √ (150*sin (25 °)/4) ≈ √ (150*0.423/4) ≈ √15.8625 ≈ 3.983 cm.

3. If lengths of both diagonals of a rhombus are known (b and c), for calculation of radius of the circle entered in such parallelogram find a ratio between the work of lengths of the parties and a square root from the sum of their lengths squared: r=b*c / √ (b²+c²). For example, if diagonals have length of 10 and 15 cm, then the radius of an inscribed circle will be 10*15 / √ (10²+15²) = 150 / √ (100+225) = 150 / √ 325 ≈ 150/18.028 ≈ 8.32 cm.

4. If length of only one diagonal of a rhombus (b) and also corner size (α) in tops which are connected by this diagonal is known, then for calculation of radius of an inscribed circle multiply a half of length of diagonal by a sine of a half of the known corner: r=b*sin(α/2)/2. For example, if length of diagonal is equal to 20 cm, and corner size - 35 °, then radius will pay off so: 20*sin (35 °/2)/2 ≈ 10*0.301 ≈ 3.01 cm.

5. If all corners are equal in tops of a rhombus, then the radius of an inscribed circle will always be a half of length of the party of this figure. As the sum of corners of a quadrangle is equal in Euclidean geometry 360 °, each corner will be equal 90 °, and such special case of a rhombus will be a square.

Author: «MirrorInfo» Dream Team


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