Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 120.32   b = 204.61   c = 179.9550058537

Area: T = 10753.57215001
Perimeter: p = 504.8880058537
Semiperimeter: s = 252.4440029269

Angle ∠ A = α = 35.74110171069° = 35°44'28″ = 0.62437984265 rad
Angle ∠ B = β = 83.37879828931° = 83°22'41″ = 1.45552203252 rad
Angle ∠ C = γ = 60.881° = 60°52'52″ = 1.06325739019 rad

Height: ha = 178.7549526264
Height: hb = 105.1132863497
Height: hc = 119.5177288157

Median: ma = 183.0422105085
Median: mb = 113.8565829709
Median: mc = 141.6888289418

Inradius: r = 42.59985194631
Circumradius: R = 102.9922109257

Vertex coordinates: A[179.9550058537; 0] B[0; 0] C[13.87551659994; 119.5177288157]
Centroid: CG[64.60884081789; 39.83990960525]
Coordinates of the circumscribed circle: U[89.97550292687; 50.11985462409]
Coordinates of the inscribed circle: I[47.83300292687; 42.59985194631]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.2598982893° = 144°15'32″ = 0.62437984265 rad
∠ B' = β' = 96.62220171069° = 96°37'19″ = 1.45552203252 rad
∠ C' = γ' = 119.119° = 119°7'8″ = 1.06325739019 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 = 120.32 ; ; b = 204.61 ; ; gamma = 60° 52'52" ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 120.32**2+204.61**2 - 2 * 120.32 * 204.61 * cos(60° 52'52") } ; ; c = 179.95 ; ;


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 = 120.32 ; ; b = 204.61 ; ; c = 179.95 ; ;

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

p = a+b+c = 120.32+204.61+179.95 = 504.88 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 504.88 }{ 2 } = 252.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 252.44 * (252.44-120.32)(252.44-204.61)(252.44-179.95) } ; ; T = sqrt{ 115639300.01 } = 10753.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10753.57 }{ 120.32 } = 178.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10753.57 }{ 204.61 } = 105.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10753.57 }{ 179.95 } = 119.52 ; ;

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{ 120.32**2-204.61**2-179.95**2 }{ 2 * 204.61 * 179.95 } ) = 35° 44'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 204.61**2-120.32**2-179.95**2 }{ 2 * 120.32 * 179.95 } ) = 83° 22'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 179.95**2-120.32**2-204.61**2 }{ 2 * 204.61 * 120.32 } ) = 60° 52'52" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10753.57 }{ 252.44 } = 42.6 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120.32 }{ 2 * sin 35° 44'28" } = 102.99 ; ;




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