Triangle calculator SAS

Obtuse scalene triangle.

Sides: a = 33 b = 37 c = 60.65547607365

Area: T = 528.709850901
Perimeter: p = 130.6554760737
Semiperimeter: s = 65.32773803682

Angle ∠ A = α = 28.11104148663° = 28°6'37″ = 0.49106192935 rad
Angle ∠ B = β = 31.89895851337° = 31°53'23″ = 0.55765782577 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: h_a = 32.043293994
Height: h_b = 28.57988383249
Height: h_c = 17.43333721736

Median: m_a = 47.45326079368
Median: m_b = 45.18657278352
Median: m_c = 17.58655053951

Inradius: r = 8.09332146066
Circumradius: R = 35.01990424388

Vertex coordinates: A[60.65547607365; 0] B[0; 0] C[28.01992350834; 17.43333721736]
Centroid: CG[29.55879986066; 5.81111240579]
Coordinates of the circumscribed circle: U[30.32773803682; -17.51095212194]
Coordinates of the inscribed circle: I[28.32773803682; 8.09332146066]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.8989585134° = 151°53'23″ = 0.49106192935 rad
∠ B' = β' = 148.1110414866° = 148°6'37″ = 0.55765782577 rad
∠ C' = γ' = 60° = 2.09443951024 rad

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How did we calculate this triangle?

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

$a = 33 ; ; b = 37 ; ; gamma = 120° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 33**2+37**2 - 2 * 33 * 37 * cos 120° } ; ; c = 60.65 ; ;$

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.

$a = 33 ; ; b = 37 ; ; c = 60.65 ; ;$

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

$p = a+b+c = 33+37+60.65 = 130.65 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 130.65 }{ 2 } = 65.33 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.33 * (65.33-33)(65.33-37)(65.33-60.65) } ; ; T = sqrt{ 279532.69 } = 528.71 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 528.71 }{ 33 } = 32.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 528.71 }{ 37 } = 28.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 528.71 }{ 60.65 } = 17.43 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 37**2+60.65**2-33**2 }{ 2 * 37 * 60.65 } ) = 28° 6'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 33**2+60.65**2-37**2 }{ 2 * 33 * 60.65 } ) = 31° 53'23" ; ; gamma = 180° - alpha - beta = 180° - 28° 6'37" - 31° 53'23" = 120° ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 528.71 }{ 65.33 } = 8.09 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 33 }{ 2 * sin 28° 6'37" } = 35.02 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 37**2+2 * 60.65**2 - 33**2 } }{ 2 } = 47.453 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 60.65**2+2 * 33**2 - 37**2 } }{ 2 } = 45.186 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 37**2+2 * 33**2 - 60.65**2 } }{ 2 } = 17.586 ; ;$
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