Triangle calculator

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

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

Right scalene triangle.

Sides: a = 2.03304167878   b = 97.5   c = 97.52111392075

Area: T = 98.98328184065
Perimeter: p = 197.0521555995
Semiperimeter: s = 98.52657779977

Angle ∠ A = α = 1.193° = 1°11'35″ = 0.0210821778 rad
Angle ∠ B = β = 88.807° = 88°48'25″ = 1.55499745488 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 97.5
Height: hb = 2.03304167878
Height: hc = 2.03299766637

Median: ma = 97.50552852315
Median: mb = 48.79222646772
Median: mc = 48.76105696038

Inradius: r = 1.00546387902
Circumradius: R = 48.76105696037

Vertex coordinates: A[97.52111392075; 0] B[0; 0] C[0.04222738328; 2.03299766637]
Centroid: CG[32.52111376801; 0.67766588879]
Coordinates of the circumscribed circle: U[48.76105696038; -0]
Coordinates of the inscribed circle: I[1.02657779977; 1.00546387902]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 178.807° = 178°48'25″ = 0.0210821778 rad
∠ B' = β' = 91.193° = 91°11'35″ = 1.55499745488 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 and angle γ.

b = 97.5 ; ; alpha = 1.193° ; ; gamma = 90° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 1.193 ° - 90 ° = 88.807 ° ; ;

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 = 97.5 * fraction{ sin 1° 11'35" }{ sin 88° 48'25" } = 2.03 ; ;

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{ 97.5**2+2.03**2 - 2 * 97.5 * 2.03 * cos(90° ) } ; ; c = 97.52 ; ;


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 = 2.03 ; ; b = 97.5 ; ; c = 97.52 ; ;

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

p = a+b+c = 2.03+97.5+97.52 = 197.05 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 197.05 }{ 2 } = 98.53 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 98.53 * (98.53-2.03)(98.53-97.5)(98.53-97.52) } ; ; T = sqrt{ 9797.6 } = 98.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 98.98 }{ 2.03 } = 97.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 98.98 }{ 97.5 } = 2.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 98.98 }{ 97.52 } = 2.03 ; ;

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{ 97.5**2+97.52**2-2.03**2 }{ 2 * 97.5 * 97.52 } ) = 1° 11'35" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.03**2+97.52**2-97.5**2 }{ 2 * 2.03 * 97.52 } ) = 88° 48'25" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 2.03**2+97.5**2-97.52**2 }{ 2 * 2.03 * 97.5 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 98.98 }{ 98.53 } = 1 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.03 }{ 2 * sin 1° 11'35" } = 48.76 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 97.5**2+2 * 97.52**2 - 2.03**2 } }{ 2 } = 97.505 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 97.52**2+2 * 2.03**2 - 97.5**2 } }{ 2 } = 48.792 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 97.5**2+2 * 2.03**2 - 97.52**2 } }{ 2 } = 48.761 ; ;
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