Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 13.1   b = 4.5   c = 9.88443752137

Area: T = 17.73884978074
Perimeter: p = 27.48443752137
Semiperimeter: s = 13.74221876069

Angle ∠ A = α = 127.0988481784° = 127°5'55″ = 2.21882869814 rad
Angle ∠ B = β = 15.90215182158° = 15°54'5″ = 0.27875338489 rad
Angle ∠ C = γ = 37° = 0.64657718232 rad

Height: ha = 2.70881676042
Height: hb = 7.88437768033
Height: hc = 3.58991996052

Median: ma = 4.00991067188
Median: mb = 11.38438893478
Median: mc = 8.45660500033

Inradius: r = 1.29108059703
Circumradius: R = 8.21221373125

Vertex coordinates: A[9.88443752137; 0] B[0; 0] C[12.59987160534; 3.58991996052]
Centroid: CG[7.49443637557; 1.19663998684]
Coordinates of the circumscribed circle: U[4.94221876069; 6.55985044712]
Coordinates of the inscribed circle: I[9.24221876069; 1.29108059703]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 52.90215182158° = 52°54'5″ = 2.21882869814 rad
∠ B' = β' = 164.0988481784° = 164°5'55″ = 0.27875338489 rad
∠ C' = γ' = 143° = 0.64657718232 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 = 13.1 ; ; b = 4.5 ; ; gamma = 37° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 13.1**2+4.5**2 - 2 * 13.1 * 4.5 * cos(37° ) } ; ; c = 9.88 ; ;


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 = 13.1 ; ; b = 4.5 ; ; c = 9.88 ; ;

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

p = a+b+c = 13.1+4.5+9.88 = 27.48 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.48 }{ 2 } = 13.74 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.74 * (13.74-13.1)(13.74-4.5)(13.74-9.88) } ; ; T = sqrt{ 314.65 } = 17.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.74 }{ 13.1 } = 2.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.74 }{ 4.5 } = 7.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.74 }{ 9.88 } = 3.59 ; ;

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{ 13.1**2-4.5**2-9.88**2 }{ 2 * 4.5 * 9.88 } ) = 127° 5'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.5**2-13.1**2-9.88**2 }{ 2 * 13.1 * 9.88 } ) = 15° 54'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.88**2-13.1**2-4.5**2 }{ 2 * 4.5 * 13.1 } ) = 37° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.74 }{ 13.74 } = 1.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13.1 }{ 2 * sin 127° 5'55" } = 8.21 ; ;




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