Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 141   b = 145   c = 49.81993610564

Area: T = 3496.301091515
Perimeter: p = 335.8199361056
Semiperimeter: s = 167.9109680528

Angle ∠ A = α = 75.46548723754° = 75°27'54″ = 1.31771104925 rad
Angle ∠ B = β = 84.53551276246° = 84°32'6″ = 1.47554163106 rad
Angle ∠ C = γ = 20° = 0.34990658504 rad

Height: ha = 49.59329207822
Height: hb = 48.22548402089
Height: hc = 140.359912308

Median: ma = 82.36603931998
Median: mb = 76.97655439606
Median: mc = 140.8287936916

Inradius: r = 20.82325094834
Circumradius: R = 72.8311033535

Vertex coordinates: A[49.81993610564; 0] B[0; 0] C[13.42882004796; 140.359912308]
Centroid: CG[21.0832520512; 46.78663743601]
Coordinates of the circumscribed circle: U[24.91096805282; 68.4398784777]
Coordinates of the inscribed circle: I[22.91096805282; 20.82325094834]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 104.5355127625° = 104°32'6″ = 1.31771104925 rad
∠ B' = β' = 95.46548723754° = 95°27'54″ = 1.47554163106 rad
∠ C' = γ' = 160° = 0.34990658504 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 = 141 ; ; b = 145 ; ; gamma = 20° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 141**2+145**2 - 2 * 141 * 145 * cos(20° ) } ; ; c = 49.82 ; ;


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 = 141 ; ; b = 145 ; ; c = 49.82 ; ;

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

p = a+b+c = 141+145+49.82 = 335.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 335.82 }{ 2 } = 167.91 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 167.91 * (167.91-141)(167.91-145)(167.91-49.82) } ; ; T = sqrt{ 12224120.09 } = 3496.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3496.3 }{ 141 } = 49.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3496.3 }{ 145 } = 48.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3496.3 }{ 49.82 } = 140.36 ; ;

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{ 141**2-145**2-49.82**2 }{ 2 * 145 * 49.82 } ) = 75° 27'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 145**2-141**2-49.82**2 }{ 2 * 141 * 49.82 } ) = 84° 32'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 49.82**2-141**2-145**2 }{ 2 * 145 * 141 } ) = 20° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3496.3 }{ 167.91 } = 20.82 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 141 }{ 2 * sin 75° 27'54" } = 72.83 ; ;




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