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?

1. Use Law of Cosines

a = 48.8 ; ; b = 39.9 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 39.9**2 = 48.8**2 + c**2 -2 * 48.8 * c * cos (22° ) ; ; ; ; c**2 -90.493c +789.43 =0 ; ; p=1; q=-90.493; 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} = 45.2465721 ± 35.4657903769 ; ; c_{1} = 80.7123624769 ; ;
c_{2} = 9.78078172309 ; ; ; ; text{ Factored form: } ; ; (c -80.7123624769) (c -9.78078172309) = 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 = 80.71 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 39.9**2+80.71**2-48.8**2 }{ 2 * 39.9 * 80.71 } ) = 27° 16'8" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 48.8**2+80.71**2-39.9**2 }{ 2 * 48.8 * 80.71 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 27° 16'8" - 22° = 130° 43'52" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.9**2+2 * 80.71**2 - 48.8**2 } }{ 2 } = 58.804 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 80.71**2+2 * 48.8**2 - 39.9**2 } }{ 2 } = 63.639 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.9**2+2 * 48.8**2 - 80.71**2 } }{ 2 } = 18.924 ; ;





#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

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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 - 2ac cos beta ; ; 39.9**2 = 48.8**2 + c**2 -2 * 48.8 * c * cos (22° ) ; ; ; ; c**2 -90.493c +789.43 =0 ; ; p=1; q=-90.493; 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} = 45.2465721 ± 35.4657903769 ; ; c_{1} = 80.7123624769 ; ; : Nr. 1
c_{2} = 9.78078172309 ; ; ; ; text{ Factored form: } ; ; (c -80.7123624769) (c -9.78078172309) = 0 ; ; ; ; c>0 ; ; : Nr. 1
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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 39.9**2+9.78**2-48.8**2 }{ 2 * 39.9 * 9.78 } ) = 152° 43'52" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 48.8**2+9.78**2-39.9**2 }{ 2 * 48.8 * 9.78 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 152° 43'52" - 22° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.9**2+2 * 9.78**2 - 48.8**2 } }{ 2 } = 15.763 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.78**2+2 * 48.8**2 - 39.9**2 } }{ 2 } = 28.992 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.9**2+2 * 48.8**2 - 9.78**2 } }{ 2 } = 44.304 ; ;
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