Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Obtuse scalene triangle.

Sides: a = 3.6   b = 2.5   c = 5.17985257276

Area: T = 4.07883850417
Perimeter: p = 11.27985257276
Semiperimeter: s = 5.63992628638

Angle ∠ A = α = 39.05334966494° = 39°3'13″ = 0.68216121009 rad
Angle ∠ B = β = 25.94765033506° = 25°56'47″ = 0.45328519128 rad
Angle ∠ C = γ = 115° = 2.00771286398 rad

Height: ha = 2.26657694676
Height: hb = 3.26327080333
Height: hc = 1.57551143303

Median: ma = 3.64660340585
Median: mb = 4.2810895275
Median: mc = 1.70331493834

Inradius: r = 0.72332124375
Circumradius: R = 2.85769354703

Vertex coordinates: A[5.17985257276; 0] B[0; 0] C[3.23771306502; 1.57551143303]
Centroid: CG[2.80552187926; 0.52550381101]
Coordinates of the circumscribed circle: U[2.58992628638; -1.20773931024]
Coordinates of the inscribed circle: I[3.13992628638; 0.72332124375]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.9476503351° = 140°56'47″ = 0.68216121009 rad
∠ B' = β' = 154.0533496649° = 154°3'13″ = 0.45328519128 rad
∠ C' = γ' = 65° = 2.00771286398 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 = 3.6 ; ; b = 2.5 ; ; gamma = 115° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 3.6**2+2.5**2 - 2 * 3.6 * 2.5 * cos 115° } ; ; c = 5.18 ; ;


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 = 3.6 ; ; b = 2.5 ; ; c = 5.18 ; ;

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

p = a+b+c = 3.6+2.5+5.18 = 11.28 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.28 }{ 2 } = 5.64 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.64 * (5.64-3.6)(5.64-2.5)(5.64-5.18) } ; ; T = sqrt{ 16.63 } = 4.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.08 }{ 3.6 } = 2.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.08 }{ 2.5 } = 3.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.08 }{ 5.18 } = 1.58 ; ;

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{ 2.5**2+5.18**2-3.6**2 }{ 2 * 2.5 * 5.18 } ) = 39° 3'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.6**2+5.18**2-2.5**2 }{ 2 * 3.6 * 5.18 } ) = 25° 56'47" ; ; gamma = 180° - alpha - beta = 180° - 39° 3'13" - 25° 56'47" = 115° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.08 }{ 5.64 } = 0.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.6 }{ 2 * sin 39° 3'13" } = 2.86 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 5.18**2 - 3.6**2 } }{ 2 } = 3.646 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.18**2+2 * 3.6**2 - 2.5**2 } }{ 2 } = 4.281 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 3.6**2 - 5.18**2 } }{ 2 } = 1.703 ; ;
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