Triangle calculator

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

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

Right scalene triangle.

Sides: a = 890.9433320909   b = 170   c = 874.5744182715

Area: T = 74338.80655308
Perimeter: p = 1935.518750362
Semiperimeter: s = 967.7598751812

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 11° = 0.19219862177 rad
Angle ∠ C = γ = 79° = 1.37988101091 rad

Height: ha = 166.8776621186
Height: hb = 874.5744182715
Height: hc = 170

Median: ma = 445.4721660454
Median: mb = 878.6955055791
Median: mc = 469.1699479259

Inradius: r = 76.81554309032
Circumradius: R = 445.4721660454

Vertex coordinates: A[874.5744182715; 0] B[0; 0] C[874.5744182715; 170]
Centroid: CG[583.0499455143; 56.66766666667]
Coordinates of the circumscribed circle: U[437.2877091358; 85]
Coordinates of the inscribed circle: I[797.7598751812; 76.81554309032]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 169° = 0.19219862177 rad
∠ C' = γ' = 101° = 1.37988101091 rad

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How did we calculate this triangle?

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

b = 170 ; ; alpha = 90° ; ; beta = 11° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 11 ° = 79 ° ; ;

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

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 170 * fraction{ sin 90° }{ sin 11° } = 890.94 ; ;

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

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 170**2+890.94**2 - 2 * 170 * 890.94 * cos(79° ) } ; ; c = 874.57 ; ;


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 = 890.94 ; ; b = 170 ; ; c = 874.57 ; ;

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

p = a+b+c = 890.94+170+874.57 = 1935.52 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1935.52 }{ 2 } = 967.76 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 967.76 * (967.76-890.94)(967.76-170)(967.76-874.57) } ; ; T = sqrt{ 5526258007.74 } = 74338.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 74338.81 }{ 890.94 } = 166.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 74338.81 }{ 170 } = 874.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 74338.81 }{ 874.57 } = 170 ; ;

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{ 170**2+874.57**2-890.94**2 }{ 2 * 170 * 874.57 } ) = 90° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 890.94**2+874.57**2-170**2 }{ 2 * 890.94 * 874.57 } ) = 11° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 890.94**2+170**2-874.57**2 }{ 2 * 890.94 * 170 } ) = 79° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 74338.81 }{ 967.76 } = 76.82 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 890.94 }{ 2 * sin 90° } = 445.47 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 170**2+2 * 874.57**2 - 890.94**2 } }{ 2 } = 445.472 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 874.57**2+2 * 890.94**2 - 170**2 } }{ 2 } = 878.695 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 170**2+2 * 890.94**2 - 874.57**2 } }{ 2 } = 469.169 ; ;
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