Triangle calculator

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

You have entered side b, height hc, angle α and angle γ.

Right scalene triangle.

Sides: a = 55.89325580794   b = 111.8   c = 124.9932871991

Area: T = 3124.394399664
Perimeter: p = 292.6855430071
Semiperimeter: s = 146.3432715035

Angle ∠ A = α = 26.562° = 26°33'43″ = 0.46435943559 rad
Angle ∠ B = β = 63.438° = 63°26'17″ = 1.10772019709 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 111.8
Height: hb = 55.89325580794
Height: hc = 49.99331547593

Median: ma = 115.2439899827
Median: mb = 79.04992760793
Median: mc = 62.49664359957

Inradius: r = 21.3549843044
Circumradius: R = 62.49664359957

Vertex coordinates: A[124.9932871991; 0] B[0; 0] C[24.9933249606; 49.99331547593]
Centroid: CG[49.99553738658; 16.66443849198]
Coordinates of the circumscribed circle: U[62.49664359957; 0]
Coordinates of the inscribed circle: I[34.54327150354; 21.3549843044]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.438° = 153°26'17″ = 0.46435943559 rad
∠ B' = β' = 116.562° = 116°33'43″ = 1.10772019709 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side b, angle α, angle γ and height hc.

b = 111.8 ; ; alpha = 26.562° ; ; gamma = 90° ; ; hc = 50 ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 26.562 ° - 90 ° = 63.438 ° ; ;

3. From angle α, angle β and side b we calculate 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 = 111.8 * fraction{ sin(26° 33'43") }{ sin (63° 26'17") } = 55.89 ; ;

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{ 111.8**2+55.89**2 - 2 * 111.8 * 55.89 * cos(90° ) } ; ; c = 124.99 ; ;


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 = 55.89 ; ; b = 111.8 ; ; c = 124.99 ; ;

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

p = a+b+c = 55.89+111.8+124.99 = 292.69 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 292.69 }{ 2 } = 146.34 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 146.34 * (146.34-55.89)(146.34-111.8)(146.34-124.99) } ; ; T = sqrt{ 9761837.85 } = 3124.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3124.39 }{ 55.89 } = 111.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3124.39 }{ 111.8 } = 55.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3124.39 }{ 124.99 } = 49.99 ; ;

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{ 55.89**2-111.8**2-124.99**2 }{ 2 * 111.8 * 124.99 } ) = 26° 33'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 111.8**2-55.89**2-124.99**2 }{ 2 * 55.89 * 124.99 } ) = 63° 26'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 124.99**2-55.89**2-111.8**2 }{ 2 * 111.8 * 55.89 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3124.39 }{ 146.34 } = 21.35 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 55.89 }{ 2 * sin 26° 33'43" } = 62.5 ; ;




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