Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 119   b = 103   c = 61.29876409871

Area: T = 3156.411084209
Perimeter: p = 283.2987640987
Semiperimeter: s = 141.6498820494

Angle ∠ A = α = 89.06879635101° = 89°4'5″ = 1.55545292213 rad
Angle ∠ B = β = 59.93220364899° = 59°55'55″ = 1.04660113642 rad
Angle ∠ C = γ = 31° = 0.54110520681 rad

Height: ha = 53.04989217157
Height: hb = 61.29895309143
Height: hc = 102.9866372436

Median: ma = 60.35768587262
Median: mb = 79.41663106376
Median: mc = 106.9844343725

Inradius: r = 22.28333542213
Circumradius: R = 59.5087873275

Vertex coordinates: A[61.29876409871; 0] B[0; 0] C[59.62222030153; 102.9866372436]
Centroid: CG[40.30766146675; 34.32987908121]
Coordinates of the circumscribed circle: U[30.64988204936; 51.00882031057]
Coordinates of the inscribed circle: I[38.64988204936; 22.28333542213]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90.93220364899° = 90°55'55″ = 1.55545292213 rad
∠ B' = β' = 120.068796351° = 120°4'5″ = 1.04660113642 rad
∠ C' = γ' = 149° = 0.54110520681 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 = 119 ; ; b = 103 ; ; gamma = 31° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 119**2+103**2 - 2 * 119 * 103 * cos(31° ) } ; ; c = 61.3 ; ;


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 = 119 ; ; b = 103 ; ; c = 61.3 ; ;

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

p = a+b+c = 119+103+61.3 = 283.3 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 283.3 }{ 2 } = 141.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 141.65 * (141.65-119)(141.65-103)(141.65-61.3) } ; ; T = sqrt{ 9962929.4 } = 3156.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3156.41 }{ 119 } = 53.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3156.41 }{ 103 } = 61.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3156.41 }{ 61.3 } = 102.99 ; ;

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{ 119**2-103**2-61.3**2 }{ 2 * 103 * 61.3 } ) = 89° 4'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 103**2-119**2-61.3**2 }{ 2 * 119 * 61.3 } ) = 59° 55'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 61.3**2-119**2-103**2 }{ 2 * 103 * 119 } ) = 31° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3156.41 }{ 141.65 } = 22.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 119 }{ 2 * sin 89° 4'5" } = 59.51 ; ;




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