Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 250.06   b = 184.35   c = 279.4777090103

Area: T = 22585.21990379
Perimeter: p = 713.8877090103
Semiperimeter: s = 356.9443545051

Angle ∠ A = α = 61.25501652914° = 61°15'1″ = 1.06990170517 rad
Angle ∠ B = β = 40.26765347086° = 40°16' = 0.7032783609 rad
Angle ∠ C = γ = 78.48333° = 78°29' = 1.37697919928 rad

Height: ha = 180.6388399088
Height: hb = 245.02554303
Height: hc = 161.6254833217

Median: ma = 201.0321545525
Median: mb = 248.6411294079
Median: mc = 169.5011038572

Inradius: r = 63.27439248294
Circumradius: R = 142.6109771291

Vertex coordinates: A[279.4777090103; 0] B[0; 0] C[190.807727682; 161.6254833217]
Centroid: CG[156.7611455641; 53.87549444057]
Coordinates of the circumscribed circle: U[139.7398545051; 28.47325463305]
Coordinates of the inscribed circle: I[172.5943545051; 63.27439248294]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.7549834709° = 118°44'59″ = 1.06990170517 rad
∠ B' = β' = 139.7333465291° = 139°44' = 0.7032783609 rad
∠ C' = γ' = 101.51767° = 101°31' = 1.37697919928 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 = 250.06 ; ; b = 184.35 ; ; gamma = 78° 29' ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 250.06**2+184.35**2 - 2 * 250.06 * 184.35 * cos(78° 29') } ; ; c = 279.48 ; ;


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 = 250.06 ; ; b = 184.35 ; ; c = 279.48 ; ;

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

p = a+b+c = 250.06+184.35+279.48 = 713.89 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 713.89 }{ 2 } = 356.94 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 356.94 * (356.94-250.06)(356.94-184.35)(356.94-279.48) } ; ; T = sqrt{ 510092118.99 } = 22585.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22585.22 }{ 250.06 } = 180.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22585.22 }{ 184.35 } = 245.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22585.22 }{ 279.48 } = 161.62 ; ;

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{ 250.06**2-184.35**2-279.48**2 }{ 2 * 184.35 * 279.48 } ) = 61° 15'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 184.35**2-250.06**2-279.48**2 }{ 2 * 250.06 * 279.48 } ) = 40° 16' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 279.48**2-250.06**2-184.35**2 }{ 2 * 184.35 * 250.06 } ) = 78° 29' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22585.22 }{ 356.94 } = 63.27 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 250.06 }{ 2 * sin 61° 15'1" } = 142.61 ; ;




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