Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 11.4   b = 13.7   c = 16.11438194112

Area: T = 76.73222332821
Perimeter: p = 41.21438194112
Semiperimeter: s = 20.60769097056

Angle ∠ A = α = 44.04404788703° = 44°2'26″ = 0.76986513604 rad
Angle ∠ B = β = 56.66595211297° = 56°39'34″ = 0.98988951963 rad
Angle ∠ C = γ = 79.3° = 79°18' = 1.38440460968 rad

Height: ha = 13.46217953126
Height: hb = 11.20217858806
Height: hc = 9.52437797227

Median: ma = 13.82768791855
Median: mb = 12.16108012898
Median: mc = 9.69107794318

Inradius: r = 3.72436167081
Circumradius: R = 8.19994756571

Vertex coordinates: A[16.11438194112; 0] B[0; 0] C[6.26655901393; 9.52437797227]
Centroid: CG[7.46598031835; 3.17545932409]
Coordinates of the circumscribed circle: U[8.05769097056; 1.52223688932]
Coordinates of the inscribed circle: I[6.90769097056; 3.72436167081]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.965952113° = 135°57'34″ = 0.76986513604 rad
∠ B' = β' = 123.344047887° = 123°20'26″ = 0.98988951963 rad
∠ C' = γ' = 100.7° = 100°42' = 1.38440460968 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 = 11.4 ; ; b = 13.7 ; ; gamma = 79° 18' ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 11.4**2+13.7**2 - 2 * 11.4 * 13.7 * cos(79° 18') } ; ; c = 16.11 ; ;


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 = 11.4 ; ; b = 13.7 ; ; c = 16.11 ; ;

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

p = a+b+c = 11.4+13.7+16.11 = 41.21 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41.21 }{ 2 } = 20.61 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.61 * (20.61-11.4)(20.61-13.7)(20.61-16.11) } ; ; T = sqrt{ 5887.84 } = 76.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 76.73 }{ 11.4 } = 13.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 76.73 }{ 13.7 } = 11.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 76.73 }{ 16.11 } = 9.52 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 11.4**2-13.7**2-16.11**2 }{ 2 * 13.7 * 16.11 } ) = 44° 2'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13.7**2-11.4**2-16.11**2 }{ 2 * 11.4 * 16.11 } ) = 56° 39'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16.11**2-11.4**2-13.7**2 }{ 2 * 13.7 * 11.4 } ) = 79° 18' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 76.73 }{ 20.61 } = 3.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.4 }{ 2 * sin 44° 2'26" } = 8.2 ; ;




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