Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 105   b = 94   c = 128.1921984823

Area: T = 4860.026626112
Perimeter: p = 327.1921984823
Semiperimeter: s = 163.5965992411

Angle ∠ A = α = 53.76989611794° = 53°46'8″ = 0.9388445408 rad
Angle ∠ B = β = 46.23110388206° = 46°13'52″ = 0.8076883844 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 92.57219287831
Height: hb = 103.4054814066
Height: hc = 75.82441830459

Median: ma = 99.39898510233
Median: mb = 107.3321693765
Median: mc = 76.3033366615

Inradius: r = 29.70774897097
Circumradius: R = 65.08547764099

Vertex coordinates: A[128.1921984823; 0] B[0; 0] C[72.63439677109; 75.82441830459]
Centroid: CG[66.9421984178; 25.2754727682]
Coordinates of the circumscribed circle: U[64.09659924115; 11.30218528174]
Coordinates of the inscribed circle: I[69.59659924115; 29.70774897097]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.2311038821° = 126°13'52″ = 0.9388445408 rad
∠ B' = β' = 133.7698961179° = 133°46'8″ = 0.8076883844 rad
∠ C' = γ' = 100° = 1.39662634016 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 = 105 ; ; b = 94 ; ; gamma = 80° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 105**2+94**2 - 2 * 105 * 94 * cos(80° ) } ; ; c = 128.19 ; ;


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 = 105 ; ; b = 94 ; ; c = 128.19 ; ;

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

p = a+b+c = 105+94+128.19 = 327.19 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 327.19 }{ 2 } = 163.6 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 163.6 * (163.6-105)(163.6-94)(163.6-128.19) } ; ; T = sqrt{ 23619855.26 } = 4860.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4860.03 }{ 105 } = 92.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4860.03 }{ 94 } = 103.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4860.03 }{ 128.19 } = 75.82 ; ;

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{ 105**2-94**2-128.19**2 }{ 2 * 94 * 128.19 } ) = 53° 46'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 94**2-105**2-128.19**2 }{ 2 * 105 * 128.19 } ) = 46° 13'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 128.19**2-105**2-94**2 }{ 2 * 94 * 105 } ) = 80° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4860.03 }{ 163.6 } = 29.71 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105 }{ 2 * sin 53° 46'8" } = 65.08 ; ;




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