Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 104.8   b = 32.3   c = 106.6659600767

Area: T = 1684.769963512
Perimeter: p = 243.7659600767
Semiperimeter: s = 121.8879800383

Angle ∠ A = α = 77.96662886646° = 77°57'59″ = 1.36107684428 rad
Angle ∠ B = β = 17.54437113354° = 17°32'37″ = 0.30661955258 rad
Angle ∠ C = γ = 84.49° = 84°29'24″ = 1.4754628685 rad

Height: ha = 32.15107563954
Height: hb = 104.316576688
Height: hc = 31.5990210783

Median: ma = 58.85659276355
Median: mb = 104.4933218526
Median: mc = 56.29547367974

Inradius: r = 13.82326320508
Circumradius: R = 53.57773569739

Vertex coordinates: A[106.6659600767; 0] B[0; 0] C[99.92554651362; 31.5990210783]
Centroid: CG[68.86216886342; 10.5330070261]
Coordinates of the circumscribed circle: U[53.33298003832; 5.14444699816]
Coordinates of the inscribed circle: I[89.58798003832; 13.82326320508]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102.0343711335° = 102°2'1″ = 1.36107684428 rad
∠ B' = β' = 162.4566288665° = 162°27'23″ = 0.30661955258 rad
∠ C' = γ' = 95.51° = 95°30'36″ = 1.4754628685 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 = 104.8 ; ; b = 32.3 ; ; gamma = 84° 29'24" ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 104.8**2+32.3**2 - 2 * 104.8 * 32.3 * cos(84° 29'24") } ; ; c = 106.66 ; ;


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 = 104.8 ; ; b = 32.3 ; ; c = 106.66 ; ;

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

p = a+b+c = 104.8+32.3+106.66 = 243.76 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 243.76 }{ 2 } = 121.88 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 121.88 * (121.88-104.8)(121.88-32.3)(121.88-106.66) } ; ; T = sqrt{ 2838212.86 } = 1684.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1684.7 }{ 104.8 } = 32.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1684.7 }{ 32.3 } = 104.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1684.7 }{ 106.66 } = 31.59 ; ;

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{ 104.8**2-32.3**2-106.66**2 }{ 2 * 32.3 * 106.66 } ) = 77° 57'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32.3**2-104.8**2-106.66**2 }{ 2 * 104.8 * 106.66 } ) = 17° 32'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 106.66**2-104.8**2-32.3**2 }{ 2 * 32.3 * 104.8 } ) = 84° 29'24" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1684.7 }{ 121.88 } = 13.82 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 104.8 }{ 2 * sin 77° 57'59" } = 53.58 ; ;




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