Triangle calculator SSA

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Triangle has two solutions with side c=80.71223624798 and with side c=9.78107817259

#1 Obtuse scalene triangle.

Sides: a = 48.8   b = 39.9   c = 80.71223624798

Area: T = 737.7433348984
Perimeter: p = 169.412236248
Semiperimeter: s = 84.70661812399

Angle ∠ A = α = 27.26987910501° = 27°16'8″ = 0.47659301869 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 130.731120895° = 130°43'52″ = 2.28216900313 rad

Height: ha = 30.23553831551
Height: hb = 36.98796164904
Height: hc = 18.28108017587

Median: ma = 58.80438070922
Median: mb = 63.63992978319
Median: mc = 18.92436263896

Inradius: r = 8.70994393607
Circumradius: R = 53.2565869893

Vertex coordinates: A[80.71223624798; 0] B[0; 0] C[45.24765721029; 18.28108017587]
Centroid: CG[41.98663115275; 6.09436005862]
Coordinates of the circumscribed circle: U[40.35661812399; -34.75500548746]
Coordinates of the inscribed circle: I[44.80661812399; 8.70994393607]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.731120895° = 152°43'52″ = 0.47659301869 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 49.26987910501° = 49°16'8″ = 2.28216900313 rad




How did we calculate this triangle?

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 = 48.8 ; ; b = 39.9 ; ; c = 80.71 ; ;

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

p = a+b+c = 48.8+39.9+80.71 = 169.41 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 169.41 }{ 2 } = 84.71 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 84.71 * (84.71-48.8)(84.71-39.9)(84.71-80.71) } ; ; T = sqrt{ 544265.25 } = 737.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 737.74 }{ 48.8 } = 30.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 737.74 }{ 39.9 } = 36.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 737.74 }{ 80.71 } = 18.28 ; ;

5. 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{ 48.8**2-39.9**2-80.71**2 }{ 2 * 39.9 * 80.71 } ) = 27° 16'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 39.9**2-48.8**2-80.71**2 }{ 2 * 48.8 * 80.71 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 80.71**2-48.8**2-39.9**2 }{ 2 * 39.9 * 48.8 } ) = 130° 43'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 737.74 }{ 84.71 } = 8.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 48.8 }{ 2 * sin 27° 16'8" } = 53.26 ; ;





#2 Obtuse scalene triangle.

Sides: a = 48.8   b = 39.9   c = 9.78107817259

Area: T = 89.44002658885
Perimeter: p = 98.48107817259
Semiperimeter: s = 49.2440390863

Angle ∠ A = α = 152.731120895° = 152°43'52″ = 2.66656624667 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 5.26987910501° = 5°16'8″ = 0.09219577514 rad

Height: ha = 3.66439453233
Height: hb = 4.48112163353
Height: hc = 18.28108017587

Median: ma = 15.76331483399
Median: mb = 28.99222290551
Median: mc = 44.30436011765

Inradius: r = 1.81655880634
Circumradius: R = 53.2565869893

Vertex coordinates: A[9.78107817259; 0] B[0; 0] C[45.24765721029; 18.28108017587]
Centroid: CG[18.34224512763; 6.09436005862]
Coordinates of the circumscribed circle: U[4.8990390863; 53.03108566333]
Coordinates of the inscribed circle: I[9.3440390863; 1.81655880634]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 27.26987910501° = 27°16'8″ = 2.66656624667 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 174.731120895° = 174°43'52″ = 0.09219577514 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 48.8 ; ; b = 39.9 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 39.9**2 = 48.8**2 + c**2 -2 * 39.9 * c * cos (22° ) ; ; ; ; c**2 -90.493c +789.43 =0 ; ; p=1; q=-90.4931442057; r=789.43 ; ; D = q**2 - 4pr = 90.493**2 - 4 * 1 * 789.43 = 5031.28914824 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 90.49 ± sqrt{ 5031.29 } }{ 2 } ; ; c_{1,2} = 45.2465721029 ± 35.4657903769 ; ;
c_{1} = 80.7123624798 ; ; c_{2} = 9.78078172594 ; ; ; ; (c -80.7123624798) (c -9.78078172594) = 0 ; ; ; ; c>0 ; ;


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 = 48.8 ; ; b = 39.9 ; ; c = 9.78 ; ;

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

p = a+b+c = 48.8+39.9+9.78 = 98.48 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 98.48 }{ 2 } = 49.24 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 49.24 * (49.24-48.8)(49.24-39.9)(49.24-9.78) } ; ; T = sqrt{ 7992.41 } = 89.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 89.4 }{ 48.8 } = 3.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 89.4 }{ 39.9 } = 4.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 89.4 }{ 9.78 } = 18.28 ; ;

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{ 48.8**2-39.9**2-9.78**2 }{ 2 * 39.9 * 9.78 } ) = 152° 43'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 39.9**2-48.8**2-9.78**2 }{ 2 * 48.8 * 9.78 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.78**2-48.8**2-39.9**2 }{ 2 * 39.9 * 48.8 } ) = 5° 16'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 89.4 }{ 49.24 } = 1.82 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 48.8 }{ 2 * sin 152° 43'52" } = 53.26 ; ;




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