Triangle calculator

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

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

Acute scalene triangle.

Sides: a = 50   b = 33.64108182402   c = 40.67436643076

Area: T = 680.4399841524
Perimeter: p = 124.3144482548
Semiperimeter: s = 62.15772412739

Angle ∠ A = α = 84° = 1.46660765717 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 54° = 0.94224777961 rad

Height: ha = 27.2165993661
Height: hb = 40.45108497187
Height: hc = 33.45765303179

Median: ma = 27.71332785869
Median: mb = 42.3598556646
Median: mc = 37.44768367674

Inradius: r = 10.94664292105
Circumradius: R = 25.13877069891

Vertex coordinates: A[40.67436643076; 0] B[0; 0] C[37.15772412739; 33.45765303179]
Centroid: CG[25.94436351938; 11.15221767726]
Coordinates of the circumscribed circle: U[20.33768321538; 14.77655734446]
Coordinates of the inscribed circle: I[28.51664230337; 10.94664292105]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 96° = 1.46660765717 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 126° = 0.94224777961 rad

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How did we calculate this triangle?

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

a = 50 ; ; alpha = 84° ; ; beta = 42° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 84 ° - 42 ° = 54 ° ; ;

3. From angle β, angle α and side a we calculate 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 = 50 * fraction{ sin(42° ) }{ sin (84° ) } = 33.64 ; ;

4. From angle γ, angle α and side a we calculate 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 = 50 * fraction{ sin(54° ) }{ sin (84° ) } = 40.67 ; ;


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 = 50 ; ; b = 33.64 ; ; c = 40.67 ; ;

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

p = a+b+c = 50+33.64+40.67 = 124.31 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 124.31 }{ 2 } = 62.16 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.16 * (62.16-50)(62.16-33.64)(62.16-40.67) } ; ; T = sqrt{ 462943.94 } = 680.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 680.4 }{ 50 } = 27.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 680.4 }{ 33.64 } = 40.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 680.4 }{ 40.67 } = 33.46 ; ;

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{ 50**2-33.64**2-40.67**2 }{ 2 * 33.64 * 40.67 } ) = 84° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 33.64**2-50**2-40.67**2 }{ 2 * 50 * 40.67 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 40.67**2-50**2-33.64**2 }{ 2 * 33.64 * 50 } ) = 54° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 680.4 }{ 62.16 } = 10.95 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 84° } = 25.14 ; ;




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