Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 150   b = 200   c = 304.1388126515

Area: T = 12990.38110568
Perimeter: p = 654.1388126515
Semiperimeter: s = 327.0699063257

Angle ∠ A = α = 25.28549960461° = 25°17'6″ = 0.44113064324 rad
Angle ∠ B = β = 34.71550039539° = 34°42'54″ = 0.60658911188 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 173.2055080757
Height: hb = 129.9043810568
Height: hc = 85.42442196177

Median: ma = 246.2211445045
Median: mb = 217.9454947177
Median: mc = 90.13987818866

Inradius: r = 39.71875475032
Circumradius: R = 175.5944229214

Vertex coordinates: A[304.1388126515; 0] B[0; 0] C[123.2999240479; 85.42442196177]
Centroid: CG[142.4799122331; 28.47547398726]
Coordinates of the circumscribed circle: U[152.0699063258; -87.79771146071]
Coordinates of the inscribed circle: I[127.0699063258; 39.71875475032]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.7155003954° = 154°42'54″ = 0.44113064324 rad
∠ B' = β' = 145.2854996046° = 145°17'6″ = 0.60658911188 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 = 150 ; ; b = 200 ; ; gamma = 120° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 150**2+200**2 - 2 * 150 * 200 * cos(120° ) } ; ; c = 304.14 ; ;


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 = 150 ; ; b = 200 ; ; c = 304.14 ; ;

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

p = a+b+c = 150+200+304.14 = 654.14 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 654.14 }{ 2 } = 327.07 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 327.07 * (327.07-150)(327.07-200)(327.07-304.14) } ; ; T = sqrt{ 168750000 } = 12990.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12990.38 }{ 150 } = 173.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12990.38 }{ 200 } = 129.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12990.38 }{ 304.14 } = 85.42 ; ;

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{ 150**2-200**2-304.14**2 }{ 2 * 200 * 304.14 } ) = 25° 17'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 200**2-150**2-304.14**2 }{ 2 * 150 * 304.14 } ) = 34° 42'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 304.14**2-150**2-200**2 }{ 2 * 200 * 150 } ) = 120° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12990.38 }{ 327.07 } = 39.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 150 }{ 2 * sin 25° 17'6" } = 175.59 ; ;




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