Triangle calculator

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

You have entered side a, angle β and angle γ.

Right scalene triangle.

Sides: a = 2000   b = 174.9777327052   c = 2007.643967509

Area: T = 174977.3277052
Perimeter: p = 4182.617700214
Semiperimeter: s = 2091.309850107

Angle ∠ A = α = 85° = 1.48435298642 rad
Angle ∠ B = β = 5° = 0.08772664626 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 174.9777327052
Height: hb = 2000
Height: hc = 174.3111485495

Median: ma = 1015.193311709
Median: mb = 2001.9132652
Median: mc = 1003.821983754

Inradius: r = 83.66988259826
Circumradius: R = 1003.821983754

Vertex coordinates: A[2007.643967509; 0] B[0; 0] C[1992.389939618; 174.3111485495]
Centroid: CG[1333.343302376; 58.10438284984]
Coordinates of the circumscribed circle: U[1003.821983754; 0]
Coordinates of the inscribed circle: I[1916.331117402; 83.66988259826]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 95° = 1.48435298642 rad
∠ B' = β' = 175° = 0.08772664626 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, angle β and angle γ.

a = 2000 ; ; beta = 5° ; ; gamma = 90° ; ;

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

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 5 ° - 90 ° = 85 ° ; ;

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

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 2000 * fraction{ sin 5° }{ sin 85° } = 174.98 ; ;

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

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 2000 * fraction{ sin 90° }{ sin 85° } = 2007.64 ; ;


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 = 2000 ; ; b = 174.98 ; ; c = 2007.64 ; ;

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

p = a+b+c = 2000+174.98+2007.64 = 4182.62 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4182.62 }{ 2 } = 2091.31 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2091.31 * (2091.31-2000)(2091.31-174.98)(2091.31-2007.64) } ; ; T = sqrt{ 30617064982.2 } = 174977.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 174977.33 }{ 2000 } = 174.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 174977.33 }{ 174.98 } = 2000 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 174977.33 }{ 2007.64 } = 174.31 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 174.98**2+2007.64**2-2000**2 }{ 2 * 174.98 * 2007.64 } ) = 85° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2000**2+2007.64**2-174.98**2 }{ 2 * 2000 * 2007.64 } ) = 5° ; ; gamma = 180° - alpha - beta = 180° - 85° - 5° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 174977.33 }{ 2091.31 } = 83.67 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2000 }{ 2 * sin 85° } = 1003.82 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 174.98**2+2 * 2007.64**2 - 2000**2 } }{ 2 } = 1015.193 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2007.64**2+2 * 2000**2 - 174.98**2 } }{ 2 } = 2001.913 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 174.98**2+2 * 2000**2 - 2007.64**2 } }{ 2 } = 1003.82 ; ;
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