Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 111.07   b = 32.25   c = 111.0698679227

Area: T = 1772.018810326
Perimeter: p = 254.3898679227
Semiperimeter: s = 127.1944339614

Angle ∠ A = α = 81.65546432843° = 81°39'17″ = 1.42551423749 rad
Angle ∠ B = β = 16.69553567157° = 16°41'43″ = 0.29113889445 rad
Angle ∠ C = γ = 81.65° = 81°39' = 1.42550613343 rad

Height: ha = 31.90881318675
Height: hb = 109.8932595551
Height: hc = 31.90985113029

Median: ma = 60.03334971298
Median: mb = 109.8932595645
Median: mc = 60.03553297957

Inradius: r = 13.93215798851
Circumradius: R = 56.12993421996

Vertex coordinates: A[111.0698679227; 0] B[0; 0] C[106.3887930737; 31.90985113029]
Centroid: CG[72.48655366546; 10.63661704343]
Coordinates of the circumscribed circle: U[55.53443396136; 8.15110845559]
Coordinates of the inscribed circle: I[94.94443396136; 13.93215798851]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98.34553567157° = 98°20'43″ = 1.42551423749 rad
∠ B' = β' = 163.3054643284° = 163°18'17″ = 0.29113889445 rad
∠ C' = γ' = 98.35° = 98°21' = 1.42550613343 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 = 111.07 ; ; b = 32.25 ; ; gamma = 81° 39' ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 111.07**2+32.25**2 - 2 * 111.07 * 32.25 * cos(81° 39') } ; ; c = 111.07 ; ;


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 = 111.07 ; ; b = 32.25 ; ; c = 111.07 ; ;

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

p = a+b+c = 111.07+32.25+111.07 = 254.39 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 254.39 }{ 2 } = 127.19 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 127.19 * (127.19-111.07)(127.19-32.25)(127.19-111.07) } ; ; T = sqrt{ 3140048.16 } = 1772.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1772.02 }{ 111.07 } = 31.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1772.02 }{ 32.25 } = 109.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1772.02 }{ 111.07 } = 31.91 ; ;

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{ 111.07**2-32.25**2-111.07**2 }{ 2 * 32.25 * 111.07 } ) = 81° 39'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.25**2-111.07**2-111.07**2 }{ 2 * 111.07 * 111.07 } ) = 16° 41'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 111.07**2-111.07**2-32.25**2 }{ 2 * 32.25 * 111.07 } ) = 81° 39' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1772.02 }{ 127.19 } = 13.93 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 111.07 }{ 2 * sin 81° 39'17" } = 56.13 ; ;




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