Triangle calculator

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

You have entered side a, b and angle γ.

Obtuse scalene triangle.

Sides: a = 10.4   b = 6.8   c = 4.01327719837

Area: T = 7.41221123843
Perimeter: p = 21.21327719837
Semiperimeter: s = 10.60663859918

Angle ∠ A = α = 147.0933234193° = 147°5'36″ = 2.56772612441 rad
Angle ∠ B = β = 20.8076765807° = 20°48'24″ = 0.363314657 rad
Angle ∠ C = γ = 12.1° = 12°6' = 0.21111848395 rad

Height: ha = 1.42554062277
Height: hb = 2.18800330542
Height: hc = 3.69442604337

Median: ma = 2.03325278587
Median: mb = 7.11113409071
Median: mc = 8.55442045365

Inradius: r = 0.69988348708
Circumradius: R = 9.57216045566

Vertex coordinates: A[4.01327719837; 0] B[0; 0] C[9.72217508633; 3.69442604337]
Centroid: CG[4.57881742823; 1.23114201446]
Coordinates of the circumscribed circle: U[2.00663859918; 9.35989544843]
Coordinates of the inscribed circle: I[3.80663859918; 0.69988348708]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 32.9076765807° = 32°54'24″ = 2.56772612441 rad
∠ B' = β' = 159.1933234193° = 159°11'36″ = 0.363314657 rad
∠ C' = γ' = 167.9° = 167°54' = 0.21111848395 rad

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How did we calculate this triangle?

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

a = 10.4 ; ; b = 6.8 ; ; gamma = 12.1° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 10.4**2+6.8**2 - 2 * 10.4 * 6.8 * cos 12° 6' } ; ; c = 4.01 ; ;
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 = 10.4 ; ; b = 6.8 ; ; c = 4.01 ; ;

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

p = a+b+c = 10.4+6.8+4.01 = 21.21 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.21 }{ 2 } = 10.61 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.61 * (10.61-10.4)(10.61-6.8)(10.61-4.01) } ; ; T = sqrt{ 54.94 } = 7.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.41 }{ 10.4 } = 1.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.41 }{ 6.8 } = 2.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.41 }{ 4.01 } = 3.69 ; ;

7. 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{ 6.8**2+4.01**2-10.4**2 }{ 2 * 6.8 * 4.01 } ) = 147° 5'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.4**2+4.01**2-6.8**2 }{ 2 * 10.4 * 4.01 } ) = 20° 48'24" ; ; gamma = 180° - alpha - beta = 180° - 147° 5'36" - 20° 48'24" = 12° 6' ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.41 }{ 10.61 } = 0.7 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.4 }{ 2 * sin 147° 5'36" } = 9.57 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.8**2+2 * 4.01**2 - 10.4**2 } }{ 2 } = 2.033 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.01**2+2 * 10.4**2 - 6.8**2 } }{ 2 } = 7.111 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.8**2+2 * 10.4**2 - 4.01**2 } }{ 2 } = 8.554 ; ;
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