Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 70   b = 52   c = 106.0387729134

Area: T = 1576.166623489
Perimeter: p = 228.0387729135
Semiperimeter: s = 114.0198864567

Angle ∠ A = α = 34.86988628107° = 34°52'8″ = 0.60985764625 rad
Angle ∠ B = β = 25.13111371893° = 25°7'52″ = 0.43986210887 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 45.03333209968
Height: hb = 60.62217782649
Height: hc = 29.72884041775

Median: ma = 75.82221603491
Median: mb = 86
Median: mc = 31.48801524774

Inradius: r = 13.82437320716
Circumradius: R = 61.22109114601

Vertex coordinates: A[106.0387729134; 0] B[0; 0] C[63.37436694934; 29.72884041775]
Centroid: CG[56.47704662093; 9.90994680592]
Coordinates of the circumscribed circle: U[53.01988645672; -30.611045573]
Coordinates of the inscribed circle: I[62.01988645672; 13.82437320716]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.1311137189° = 145°7'52″ = 0.60985764625 rad
∠ B' = β' = 154.8698862811° = 154°52'8″ = 0.43986210887 rad
∠ C' = γ' = 60° = 2.09443951024 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 = 70 ; ; b = 52 ; ; gamma = 120° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 70**2+52**2 - 2 * 70 * 52 * cos(120° ) } ; ; c = 106.04 ; ;


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 = 70 ; ; b = 52 ; ; c = 106.04 ; ;

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

p = a+b+c = 70+52+106.04 = 228.04 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 228.04 }{ 2 } = 114.02 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 114.02 * (114.02-70)(114.02-52)(114.02-106.04) } ; ; T = sqrt{ 2484300 } = 1576.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1576.17 }{ 70 } = 45.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1576.17 }{ 52 } = 60.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1576.17 }{ 106.04 } = 29.73 ; ;

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{ 70**2-52**2-106.04**2 }{ 2 * 52 * 106.04 } ) = 34° 52'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 52**2-70**2-106.04**2 }{ 2 * 70 * 106.04 } ) = 25° 7'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 106.04**2-70**2-52**2 }{ 2 * 52 * 70 } ) = 120° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1576.17 }{ 114.02 } = 13.82 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 70 }{ 2 * sin 34° 52'8" } = 61.22 ; ;




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