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.

Obtuse scalene triangle.

Sides: a = 108.7466429171   b = 43   c = 83.35109524712

Area: T = 1624.145477155
Perimeter: p = 235.0977381643
Semiperimeter: s = 117.5498690821

Angle ∠ A = α = 115° = 2.00771286398 rad
Angle ∠ B = β = 21° = 0.36765191429 rad
Angle ∠ C = γ = 44° = 0.76879448709 rad

Height: ha = 29.87703099297
Height: hb = 75.54216172815
Height: hc = 38.97112348426

Median: ma = 37.97703064846
Median: mb = 94.46986909389
Median: mc = 71.41881182146

Inradius: r = 13.81767831577
Circumradius: R = 59.99442043569

Vertex coordinates: A[83.35109524712; 0] B[0; 0] C[101.5243537726; 38.97112348426]
Centroid: CG[61.62548300657; 12.99904116142]
Coordinates of the circumscribed circle: U[41.67554762356; 43.15662189836]
Coordinates of the inscribed circle: I[74.54986908213; 13.81767831577]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 65° = 2.00771286398 rad
∠ B' = β' = 159° = 0.36765191429 rad
∠ C' = γ' = 136° = 0.76879448709 rad

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How did we calculate this triangle?

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

b = 43 ; ; alpha = 115° ; ; gamma = 44° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 115 ° - 44 ° = 21 ° ; ;

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 = 43 * fraction{ sin 115° }{ sin 21° } = 108.75 ; ;

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{ 43**2+108.75**2 - 2 * 43 * 108.75 * cos 44° } ; ; c = 83.35 ; ;
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 = 108.75 ; ; b = 43 ; ; c = 83.35 ; ;

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

p = a+b+c = 108.75+43+83.35 = 235.1 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 235.1 }{ 2 } = 117.55 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 117.55 * (117.55-108.75)(117.55-43)(117.55-83.35) } ; ; T = sqrt{ 2637846.24 } = 1624.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1624.14 }{ 108.75 } = 29.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1624.14 }{ 43 } = 75.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1624.14 }{ 83.35 } = 38.97 ; ;

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{ 43**2+83.35**2-108.75**2 }{ 2 * 43 * 83.35 } ) = 115° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 108.75**2+83.35**2-43**2 }{ 2 * 108.75 * 83.35 } ) = 21° ; ;
 gamma = 180° - alpha - beta = 180° - 115° - 21° = 44° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1624.14 }{ 117.55 } = 13.82 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 108.75 }{ 2 * sin 115° } = 59.99 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 43**2+2 * 83.35**2 - 108.75**2 } }{ 2 } = 37.97 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 83.35**2+2 * 108.75**2 - 43**2 } }{ 2 } = 94.469 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 43**2+2 * 108.75**2 - 83.35**2 } }{ 2 } = 71.418 ; ;
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