Triangle calculator SSA

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Triangle has two solutions with side c=66.35879109548 and with side c=13.94395587759

#1 Acute scalene triangle.

Sides: a = 95   b = 90   c = 66.35879109548

Area: T = 2856.683284292
Perimeter: p = 251.3587910955
Semiperimeter: s = 125.6798955477

Angle ∠ A = α = 73.06994054264° = 73°4'10″ = 1.27553017072 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 41.93105945736° = 41°55'50″ = 0.73218269326 rad

Height: ha = 60.14106914299
Height: hb = 63.48218409537
Height: hc = 86.09992397685

Median: ma = 63.20994626867
Median: mb = 68.47876326485
Median: mc = 86.38108828007

Inradius: r = 22.73300014713
Circumradius: R = 49.65220063533

Vertex coordinates: A[66.35879109548; 0] B[0; 0] C[40.14987348654; 86.09992397685]
Centroid: CG[35.50222152734; 28.76997465895]
Coordinates of the circumscribed circle: U[33.17989554774; 36.9398850122]
Coordinates of the inscribed circle: I[35.67989554774; 22.73300014713]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 106.9310594574° = 106°55'50″ = 1.27553017072 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 138.0699405426° = 138°4'10″ = 0.73218269326 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 95 ; ; b = 90 ; ; beta = 65° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 95**2 + c**2 -2 * 95 * c * cos (65° ) ; ; ; ; c**2 -80.297c +925 =0 ; ; p=1; q=-80.297; r=925 ; ; D = q**2 - 4pr = 80.297**2 - 4 * 1 * 925 = 2747.68364516 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 80.3 ± sqrt{ 2747.68 } }{ 2 } ; ; c_{1,2} = 40.14873487 ± 26.2091760895 ; ; c_{1} = 66.3579109595 ; ; c_{2} = 13.9395587805 ; ; ; ; text{ Factored form: } ; ; (c -66.3579109595) (c -13.9395587805) = 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 = 95 ; ; b = 90 ; ; c = 66.36 ; ;

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

p = a+b+c = 95+90+66.36 = 251.36 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 251.36 }{ 2 } = 125.68 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 125.68 * (125.68-95)(125.68-90)(125.68-66.36) } ; ; T = sqrt{ 8160636.87 } = 2856.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2856.68 }{ 95 } = 60.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2856.68 }{ 90 } = 63.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2856.68 }{ 66.36 } = 86.1 ; ;

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{ 90**2+66.36**2-95**2 }{ 2 * 90 * 66.36 } ) = 73° 4'10" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 95**2+66.36**2-90**2 }{ 2 * 95 * 66.36 } ) = 65° ; ; gamma = 180° - alpha - beta = 180° - 73° 4'10" - 65° = 41° 55'50" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2856.68 }{ 125.68 } = 22.73 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 95 }{ 2 * sin 73° 4'10" } = 49.65 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 66.36**2 - 95**2 } }{ 2 } = 63.209 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 66.36**2+2 * 95**2 - 90**2 } }{ 2 } = 68.478 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 95**2 - 66.36**2 } }{ 2 } = 86.381 ; ;







#2 Obtuse scalene triangle.

Sides: a = 95   b = 90   c = 13.94395587759

Area: T = 600.0932706656
Perimeter: p = 198.9439558776
Semiperimeter: s = 99.47697793879

Angle ∠ A = α = 106.9310594574° = 106°55'50″ = 1.86662909464 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 8.06994054264° = 8°4'10″ = 0.14108376934 rad

Height: ha = 12.63435306664
Height: hb = 13.33553934812
Height: hc = 86.09992397685

Median: ma = 43.48545449491
Median: mb = 50.8439508745
Median: mc = 92.27109172778

Inradius: r = 6.03329148245
Circumradius: R = 49.65220063533

Vertex coordinates: A[13.94395587759; 0] B[0; 0] C[40.14987348654; 86.09992397685]
Centroid: CG[18.02994312138; 28.76997465895]
Coordinates of the circumscribed circle: U[6.97697793879; 49.16603896465]
Coordinates of the inscribed circle: I[9.47697793879; 6.03329148245]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 73.06994054264° = 73°4'10″ = 1.86662909464 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 171.9310594574° = 171°55'50″ = 0.14108376934 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 95 ; ; b = 90 ; ; beta = 65° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 95**2 + c**2 -2 * 95 * c * cos (65° ) ; ; ; ; c**2 -80.297c +925 =0 ; ; p=1; q=-80.297; r=925 ; ; D = q**2 - 4pr = 80.297**2 - 4 * 1 * 925 = 2747.68364516 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 80.3 ± sqrt{ 2747.68 } }{ 2 } ; ; c_{1,2} = 40.14873487 ± 26.2091760895 ; ; c_{1} = 66.3579109595 ; ; c_{2} = 13.9395587805 ; ; ; ; text{ Factored form: } ; ; (c -66.3579109595) (c -13.9395587805) = 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 = 95 ; ; b = 90 ; ; c = 13.94 ; ;

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

p = a+b+c = 95+90+13.94 = 198.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 198.94 }{ 2 } = 99.47 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 99.47 * (99.47-95)(99.47-90)(99.47-13.94) } ; ; T = sqrt{ 360111.26 } = 600.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 600.09 }{ 95 } = 12.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 600.09 }{ 90 } = 13.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 600.09 }{ 13.94 } = 86.1 ; ;

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{ 90**2+13.94**2-95**2 }{ 2 * 90 * 13.94 } ) = 106° 55'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 95**2+13.94**2-90**2 }{ 2 * 95 * 13.94 } ) = 65° ; ; gamma = 180° - alpha - beta = 180° - 106° 55'50" - 65° = 8° 4'10" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 600.09 }{ 99.47 } = 6.03 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 95 }{ 2 * sin 106° 55'50" } = 49.65 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 13.94**2 - 95**2 } }{ 2 } = 43.485 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.94**2+2 * 95**2 - 90**2 } }{ 2 } = 50.84 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 95**2 - 13.94**2 } }{ 2 } = 92.271 ; ;
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