Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 500   b = 1200   c = 848.045457828

Area: T = 176335.5765688
Perimeter: p = 2548.045457828
Semiperimeter: s = 1274.022228914

Angle ∠ A = α = 20.2776642331° = 20°16'36″ = 0.35438941699 rad
Angle ∠ B = β = 123.7233357669° = 123°43'24″ = 2.15993799529 rad
Angle ∠ C = γ = 36° = 0.62883185307 rad

Height: ha = 705.3422302751
Height: hb = 293.8932626146
Height: hc = 415.8643930279

Median: ma = 1008.509870268
Median: mb = 352.9732808266
Median: mc = 815.6011065664

Inradius: r = 138.4098548415
Circumradius: R = 721.3989806032

Vertex coordinates: A[848.045457828; 0] B[0; 0] C[-277.5921771299; 415.8643930279]
Centroid: CG[190.151093566; 138.6211310093]
Coordinates of the circumscribed circle: U[424.022228914; 583.6176612649]
Coordinates of the inscribed circle: I[74.022228914; 138.4098548415]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.7233357669° = 159°43'24″ = 0.35438941699 rad
∠ B' = β' = 56.2776642331° = 56°16'36″ = 2.15993799529 rad
∠ C' = γ' = 144° = 0.62883185307 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 = 500 ; ; b = 1200 ; ; gamma = 36° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 500**2+1200**2 - 2 * 500 * 1200 * cos(36° ) } ; ; c = 848.04 ; ;


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 = 500 ; ; b = 1200 ; ; c = 848.04 ; ;

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

p = a+b+c = 500+1200+848.04 = 2548.04 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2548.04 }{ 2 } = 1274.02 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1274.02 * (1274.02-500)(1274.02-1200)(1274.02-848.04) } ; ; T = sqrt{ 31094235253.1 } = 176335.58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 176335.58 }{ 500 } = 705.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 176335.58 }{ 1200 } = 293.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 176335.58 }{ 848.04 } = 415.86 ; ;

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{ 500**2-1200**2-848.04**2 }{ 2 * 1200 * 848.04 } ) = 20° 16'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1200**2-500**2-848.04**2 }{ 2 * 500 * 848.04 } ) = 123° 43'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 848.04**2-500**2-1200**2 }{ 2 * 1200 * 500 } ) = 36° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 176335.58 }{ 1274.02 } = 138.41 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 500 }{ 2 * sin 20° 16'36" } = 721.39 ; ;




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