Triangle calculator SAS

Obtuse scalene triangle.

Sides: a = 32 b = 35 c = 58.04330874437

Area: T = 484.9744226119
Perimeter: p = 125.0433087444
Semiperimeter: s = 62.52215437219

Angle ∠ A = α = 28.51991477188° = 28°31'9″ = 0.49877530276 rad
Angle ∠ B = β = 31.48108522812° = 31°28'51″ = 0.54994445236 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: h_a = 30.31108891325
Height: h_b = 27.71328129211
Height: h_c = 16.71108349152

Median: m_a = 45.17774279923
Median: m_b = 43.47770054167
Median: m_c = 16.88002976164

Inradius: r = 7.75769138132
Circumradius: R = 33.51111921602

Vertex coordinates: A[58.04330874437; 0] B[0; 0] C[27.29900713894; 16.71108349152]
Centroid: CG[28.44443862777; 5.57702783051]
Coordinates of the circumscribed circle: U[29.02215437219; -16.75655960801]
Coordinates of the inscribed circle: I[27.52215437219; 7.75769138132]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.4810852281° = 151°28'51″ = 0.49877530276 rad
∠ B' = β' = 148.5199147719° = 148°31'9″ = 0.54994445236 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

$a = 32 ; ; b = 35 ; ; gamma = 120° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 32**2+35**2 - 2 * 32 * 35 * cos 120° } ; ; c = 58.04 ; ;$

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.

$a = 32 ; ; b = 35 ; ; c = 58.04 ; ;$

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

$p = a+b+c = 32+35+58.04 = 125.04 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 125.04 }{ 2 } = 62.52 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.52 * (62.52-32)(62.52-35)(62.52-58.04) } ; ; T = sqrt{ 235200 } = 484.97 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 484.97 }{ 32 } = 30.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 484.97 }{ 35 } = 27.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 484.97 }{ 58.04 } = 16.71 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 35**2+58.04**2-32**2 }{ 2 * 35 * 58.04 } ) = 28° 31'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 32**2+58.04**2-35**2 }{ 2 * 32 * 58.04 } ) = 31° 28'51" ; ; gamma = 180° - alpha - beta = 180° - 28° 31'9" - 31° 28'51" = 120° ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 484.97 }{ 62.52 } = 7.76 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 32 }{ 2 * sin 28° 31'9" } = 33.51 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 58.04**2 - 32**2 } }{ 2 } = 45.177 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.04**2+2 * 32**2 - 35**2 } }{ 2 } = 43.477 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 32**2 - 58.04**2 } }{ 2 } = 16.8 ; ;$
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