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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 95 ; ; b = 90 ; ; beta = 65° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 95**2 + c**2 -2 * 90 * c * cos (65° ) ; ; ; ; c**2 -80.297c +925 =0 ; ; p=1; q=-80.2974697307; 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} = 40.1487348654 ± 26.2091760895 ; ; c_{1} = 66.3579109548 ; ;
c_{2} = 13.9395587759 ; ; ; ; (c -66.3579109548) (c -13.9395587759) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 95**2-90**2-13.94**2 }{ 2 * 90 * 13.94 } ) = 106° 55'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-95**2-13.94**2 }{ 2 * 95 * 13.94 } ) = 65° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13.94**2-95**2-90**2 }{ 2 * 90 * 95 } ) = 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 ; ;




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