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.

Obtuse scalene triangle.

Sides: a = 100   b = 1.99554279894   c = 99.01773668299

Area: T = 86.40545665132
Perimeter: p = 201.0132794819
Semiperimeter: s = 100.506639741

Angle ∠ A = α = 119° = 2.07769418099 rad
Angle ∠ B = β = 1° = 0.01774532925 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 1.72880913303
Height: hb = 86.60325403784
Height: hc = 1.74552406437

Median: ma = 49.03327475611
Median: mb = 99.50548945215
Median: mc = 50.50662484545

Inradius: r = 0.86596922061
Circumradius: R = 57.16877033937

Vertex coordinates: A[99.01773668299; 0] B[0; 0] C[99.98547695156; 1.74552406437]
Centroid: CG[66.33440454485; 0.58217468812]
Coordinates of the circumscribed circle: U[49.50986834149; 28.58438516968]
Coordinates of the inscribed circle: I[98.51109694202; 0.86596922061]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 61° = 2.07769418099 rad
∠ B' = β' = 179° = 0.01774532925 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

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

a = 100 ; ; beta = 1° ; ; gamma = 60° ; ;

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

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 1 ° - 60 ° = 119 ° ; ;

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 = 100 * fraction{ sin(1° ) }{ sin (119° ) } = 2 ; ;

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 = 100 * fraction{ sin(60° ) }{ sin (119° ) } = 99.02 ; ;


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 = 100 ; ; b = 2 ; ; c = 99.02 ; ;

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

p = a+b+c = 100+2+99.02 = 201.01 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 201.01 }{ 2 } = 100.51 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.51 * (100.51-100)(100.51-2)(100.51-99.02) } ; ; T = sqrt{ 7465.75 } = 86.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 86.4 }{ 100 } = 1.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 86.4 }{ 2 } = 86.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 86.4 }{ 99.02 } = 1.75 ; ;

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{ 100**2-2**2-99.02**2 }{ 2 * 2 * 99.02 } ) = 119° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2**2-100**2-99.02**2 }{ 2 * 100 * 99.02 } ) = 1° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 99.02**2-100**2-2**2 }{ 2 * 2 * 100 } ) = 60° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 86.4 }{ 100.51 } = 0.86 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 119° } = 57.17 ; ;




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