Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 516   b = 356   c = 505.3366060886

Area: T = 85159.98326743
Perimeter: p = 1377.336606089
Semiperimeter: s = 688.6688030443

Angle ∠ A = α = 71.21879023847° = 71°13'4″ = 1.2432986883 rad
Angle ∠ B = β = 40.78220976153° = 40°46'56″ = 0.71217818793 rad
Angle ∠ C = γ = 68° = 1.18768238914 rad

Height: ha = 330.0777452226
Height: hb = 478.4276868957
Height: hc = 337.043296711

Median: ma = 352.8266114702
Median: mb = 478.6711356168
Median: mc = 364.2188157691

Inradius: r = 123.6598974876
Circumradius: R = 272.5111249197

Vertex coordinates: A[505.3366060886; 0] B[0; 0] C[390.7154778735; 337.043296711]
Centroid: CG[298.6843613207; 112.3487655703]
Coordinates of the circumscribed circle: U[252.6688030443; 102.0854510729]
Coordinates of the inscribed circle: I[332.6688030443; 123.6598974876]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.7822097615° = 108°46'56″ = 1.2432986883 rad
∠ B' = β' = 139.2187902385° = 139°13'4″ = 0.71217818793 rad
∠ C' = γ' = 112° = 1.18768238914 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 = 516 ; ; b = 356 ; ; gamma = 68° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 516**2+356**2 - 2 * 516 * 356 * cos(68° ) } ; ; c = 505.34 ; ;


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 = 516 ; ; b = 356 ; ; c = 505.34 ; ;

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

p = a+b+c = 516+356+505.34 = 1377.34 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1377.34 }{ 2 } = 688.67 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 688.67 * (688.67-516)(688.67-356)(688.67-505.34) } ; ; T = sqrt{ 7252222649.08 } = 85159.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 85159.98 }{ 516 } = 330.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 85159.98 }{ 356 } = 478.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 85159.98 }{ 505.34 } = 337.04 ; ;

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{ 516**2-356**2-505.34**2 }{ 2 * 356 * 505.34 } ) = 71° 13'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 356**2-516**2-505.34**2 }{ 2 * 516 * 505.34 } ) = 40° 46'56" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 505.34**2-516**2-356**2 }{ 2 * 356 * 516 } ) = 68° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 85159.98 }{ 688.67 } = 123.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 516 }{ 2 * sin 71° 13'4" } = 272.51 ; ;




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