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 = 70.35878624556   b = 39   c = 80.44439482455

Area: T = 1371.978831788
Perimeter: p = 189.8021810701
Semiperimeter: s = 94.90109053505

Angle ∠ A = α = 61° = 1.06546508437 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 39
Height: hb = 70.35878624556
Height: hc = 34.11101685784

Median: ma = 52.52219687591
Median: mb = 73.01101281284
Median: mc = 40.22219741227

Inradius: r = 14.45769571051
Circumradius: R = 40.22219741227

Vertex coordinates: A[80.44439482455; 0] B[0; 0] C[61.53663730559; 34.11101685784]
Centroid: CG[47.32767737671; 11.37700561928]
Coordinates of the circumscribed circle: U[40.22219741227; 0]
Coordinates of the inscribed circle: I[55.90109053505; 14.45769571051]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119° = 1.06546508437 rad
∠ B' = β' = 151° = 0.50661454831 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 = 39 ; ; alpha = 61° ; ; gamma = 90° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 61 ° - 90 ° = 29 ° ; ;

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 = 39 * fraction{ sin 61° }{ sin 29° } = 70.36 ; ;

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{ 39**2+70.36**2 - 2 * 39 * 70.36 * cos 90° } ; ; c = 80.44 ; ;
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 = 70.36 ; ; b = 39 ; ; c = 80.44 ; ;

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

p = a+b+c = 70.36+39+80.44 = 189.8 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 189.8 }{ 2 } = 94.9 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 94.9 * (94.9-70.36)(94.9-39)(94.9-80.44) } ; ; T = sqrt{ 1882324.5 } = 1371.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1371.98 }{ 70.36 } = 39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1371.98 }{ 39 } = 70.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1371.98 }{ 80.44 } = 34.11 ; ;

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{ 39**2+80.44**2-70.36**2 }{ 2 * 39 * 80.44 } ) = 61° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 70.36**2+80.44**2-39**2 }{ 2 * 70.36 * 80.44 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 61° - 29° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1371.98 }{ 94.9 } = 14.46 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 70.36 }{ 2 * sin 61° } = 40.22 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 80.44**2 - 70.36**2 } }{ 2 } = 52.522 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 80.44**2+2 * 70.36**2 - 39**2 } }{ 2 } = 73.01 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 70.36**2 - 80.44**2 } }{ 2 } = 40.222 ; ;
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