Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 1.8   b = 5.1   c = 6.01106386562

Area: T = 4.25657738925
Perimeter: p = 12.91106386562
Semiperimeter: s = 6.45553193281

Angle ∠ A = α = 16.12107644889° = 16°7'15″ = 0.28113604183 rad
Angle ∠ B = β = 51.87992355111° = 51°52'45″ = 0.90554634731 rad
Angle ∠ C = γ = 112° = 1.95547687622 rad

Height: ha = 4.72986376583
Height: hb = 1.66989309382
Height: hc = 1.4166080432

Median: ma = 5.50108079886
Median: mb = 3.63106182018
Median: mc = 2.36549642146

Inradius: r = 0.65992662076
Circumradius: R = 3.24113413082

Vertex coordinates: A[6.01106386562; 0] B[0; 0] C[1.11111778481; 1.4166080432]
Centroid: CG[2.37439388348; 0.47220268107]
Coordinates of the circumscribed circle: U[3.00553193281; -1.21442278256]
Coordinates of the inscribed circle: I[1.35553193281; 0.65992662076]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.8799235511° = 163°52'45″ = 0.28113604183 rad
∠ B' = β' = 128.1210764489° = 128°7'15″ = 0.90554634731 rad
∠ C' = γ' = 68° = 1.95547687622 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 = 1.8 ; ; b = 5.1 ; ; gamma = 112° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 1.8**2+5.1**2 - 2 * 1.8 * 5.1 * cos 112° } ; ; c = 6.01 ; ;


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 = 1.8 ; ; b = 5.1 ; ; c = 6.01 ; ;

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

p = a+b+c = 1.8+5.1+6.01 = 12.91 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.91 }{ 2 } = 6.46 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.46 * (6.46-1.8)(6.46-5.1)(6.46-6.01) } ; ; T = sqrt{ 18.11 } = 4.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.26 }{ 1.8 } = 4.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.26 }{ 5.1 } = 1.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.26 }{ 6.01 } = 1.42 ; ;

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{ 5.1**2+6.01**2-1.8**2 }{ 2 * 5.1 * 6.01 } ) = 16° 7'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.8**2+6.01**2-5.1**2 }{ 2 * 1.8 * 6.01 } ) = 51° 52'45" ; ; gamma = 180° - alpha - beta = 180° - 16° 7'15" - 51° 52'45" = 112° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.26 }{ 6.46 } = 0.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.8 }{ 2 * sin 16° 7'15" } = 3.24 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 6.01**2 - 1.8**2 } }{ 2 } = 5.501 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.01**2+2 * 1.8**2 - 5.1**2 } }{ 2 } = 3.631 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 1.8**2 - 6.01**2 } }{ 2 } = 2.365 ; ;
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