Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side c, angle α and angle β.

Right scalene triangle.

Sides: a = 4965.212231503   b = 2482.606615752   c = 4300

Area: T = 5337603.239866
Perimeter: p = 11747.81884725
Semiperimeter: s = 5873.909923627

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 2150
Height: hb = 4300
Height: hc = 2482.606615752

Median: ma = 2482.606615752
Median: mb = 4475.582189885
Median: mc = 3284.179924805

Inradius: r = 908.6976921242
Circumradius: R = 2482.606615752

Vertex coordinates: A[4300; 0] B[0; 0] C[4300; 2482.606615752]
Centroid: CG[2866.667666667; 827.5355385839]
Coordinates of the circumscribed circle: U[2150; 1241.303307876]
Coordinates of the inscribed circle: I[3391.303307876; 908.6976921242]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle α and angle β.

c = 4300 ; ; alpha = 90° ; ; beta = 30° ; ;

2. From angle α and angle β we calculate γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 30 ° = 60 ° ; ;

3. From angle α, angle γ and side c we calculate a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 4300 * fraction{ sin(90° ) }{ sin (60° ) } = 4965.21 ; ;

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 4965.21**2+4300**2 - 2 * 4965.21 * 4300 * cos(30° ) } ; ; b = 2482.61 ; ;


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 = 4965.21 ; ; b = 2482.61 ; ; c = 4300 ; ;

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

p = a+b+c = 4965.21+2482.61+4300 = 11747.82 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11747.82 }{ 2 } = 5873.91 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5873.91 * (5873.91-4965.21)(5873.91-2482.61)(5873.91-4300) } ; ; T = sqrt{ 2.849 * 10**{ 13 } } = 5337603.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5337603.24 }{ 4965.21 } = 2150 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5337603.24 }{ 2482.61 } = 4300 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5337603.24 }{ 4300 } = 2482.61 ; ;

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{ 4965.21**2-2482.61**2-4300**2 }{ 2 * 2482.61 * 4300 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2482.61**2-4965.21**2-4300**2 }{ 2 * 4965.21 * 4300 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4300**2-4965.21**2-2482.61**2 }{ 2 * 2482.61 * 4965.21 } ) = 60° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5337603.24 }{ 5873.91 } = 908.7 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4965.21 }{ 2 * sin 90° } = 2482.61 ; ;




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