Right triangle calculator (B,a)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered cathetus a and angle β.

Right scalene triangle.

Sides: a = 860   b = 347.4632554218   c = 927.5439878703

Area: T = 149408.8988314
Perimeter: p = 2135.002243292
Semiperimeter: s = 1067.501121646

Angle ∠ A = α = 68° = 1.18768238914 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 347.4632554218
Height: hb = 860
Height: hc = 322.1621670338

Median: ma = 552.8388336753
Median: mb = 877.3732530141
Median: mc = 463.7769939351

Inradius: r = 139.9611337758
Circumradius: R = 463.7769939351

Vertex coordinates: A[927.5439878703; 0] B[0; 0] C[797.3788114927; 322.1621670338]
Centroid: CG[574.9732664543; 107.3877223446]
Coordinates of the circumscribed circle: U[463.7769939351; 0]
Coordinates of the inscribed circle: I[720.0398662242; 139.9611337758]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112° = 1.18768238914 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a angle β

a = 860 ; ; beta = 22° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 22 ° = 68 ° ; ;

3. From cathetus a and angle α we calculate hypotenuse c:

 sin alpha = a:c ; ; c = a/ sin alpha = a/ sin(68 ° ) = 927.54 ; ;

4. From cathetus a and hypotenuse c we calculate cathetus b - Pythagorean theorem:

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 927.54**2 - 860**2 } = 347.463 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 860 ; ; b = 347.46 ; ; c = 927.54 ; ;

5. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 860+347.46+927.54 = 2135 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2135 }{ 2 } = 1067.5 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1067.5 * (1067.5-860)(1067.5-347.46)(1067.5-927.54) } ; ; T = sqrt{ 22323018895.4 } = 149408.9 ; ;

8. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 149408.9 }{ 860 } = 347.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 149408.9 }{ 347.46 } = 860 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 149408.9 }{ 927.54 } = 322.16 ; ;

9. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 860**2-347.46**2-927.54**2 }{ 2 * 347.46 * 927.54 } ) = 68° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 347.46**2-860**2-927.54**2 }{ 2 * 860 * 927.54 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 927.54**2-860**2-347.46**2 }{ 2 * 347.46 * 860 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 149408.9 }{ 1067.5 } = 139.96 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 860 }{ 2 * sin 68° } = 463.77 ; ;
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