Triangle calculator SSA

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Triangle has two solutions with side c=20.67219080978 and with side c=4.69223583223

#1 Acute scalene triangle.

Sides: a = 49   b = 48   c = 20.67219080978

Area: T = 489.2044482803
Perimeter: p = 117.6721908098
Semiperimeter: s = 58.83659540489

Angle ∠ A = α = 80.41883100098° = 80°25'6″ = 1.40435642886 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 24.58216899902° = 24°34'54″ = 0.4299031426 rad

Height: ha = 19.96875299103
Height: hb = 20.38435201168
Height: hc = 47.33303654882

Median: ma = 27.66661145122
Median: mb = 28.95110602949
Median: mc = 47.38884801814

Inradius: r = 8.31547199822
Circumradius: R = 24.84766283298

Vertex coordinates: A[20.67219080978; 0] B[0; 0] C[12.682213321; 47.33303654882]
Centroid: CG[11.11880137693; 15.77767884961]
Coordinates of the circumscribed circle: U[10.33659540489; 22.59547558796]
Coordinates of the inscribed circle: I[10.83659540489; 8.31547199822]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 99.58216899902° = 99°34'54″ = 1.40435642886 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 155.418831001° = 155°25'6″ = 0.4299031426 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 49 ; ; b = 48 ; ; beta = 75° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 48**2 = 49**2 + c**2 -2 * 49 * c * cos (75° ) ; ; ; ; c**2 -25.364c +97 =0 ; ; p=1; q=-25.364; r=97 ; ; D = q**2 - 4pr = 25.364**2 - 4 * 1 * 97 = 255.346011027 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 25.36 ± sqrt{ 255.35 } }{ 2 } ; ; c_{1,2} = 12.68213321 ± 7.98977488774 ; ; c_{1} = 20.6719080977 ; ;
c_{2} = 4.69235832226 ; ; ; ; text{ Factored form: } ; ; (c -20.6719080977) (c -4.69235832226) = 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 = 49 ; ; b = 48 ; ; c = 20.67 ; ;

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

p = a+b+c = 49+48+20.67 = 117.67 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 117.67 }{ 2 } = 58.84 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 58.84 * (58.84-49)(58.84-48)(58.84-20.67) } ; ; T = sqrt{ 239321.03 } = 489.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 489.2 }{ 49 } = 19.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 489.2 }{ 48 } = 20.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 489.2 }{ 20.67 } = 47.33 ; ;

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{ 48**2+20.67**2-49**2 }{ 2 * 48 * 20.67 } ) = 80° 25'6" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 49**2+20.67**2-48**2 }{ 2 * 49 * 20.67 } ) = 75° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 49**2+48**2-20.67**2 }{ 2 * 49 * 48 } ) = 24° 34'54" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 489.2 }{ 58.84 } = 8.31 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 49 }{ 2 * sin 80° 25'6" } = 24.85 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 20.67**2 - 49**2 } }{ 2 } = 27.666 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.67**2+2 * 49**2 - 48**2 } }{ 2 } = 28.951 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 49**2 - 20.67**2 } }{ 2 } = 47.388 ; ;







#2 Obtuse scalene triangle.

Sides: a = 49   b = 48   c = 4.69223583223

Area: T = 111.0465517198
Perimeter: p = 101.6922358322
Semiperimeter: s = 50.84661791611

Angle ∠ A = α = 99.58216899902° = 99°34'54″ = 1.7388028365 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 5.41883100098° = 5°25'6″ = 0.09545673496 rad

Height: ha = 4.53224700897
Height: hb = 4.62768965499
Height: hc = 47.33303654882

Median: ma = 23.72325444106
Median: mb = 25.20993060855
Median: mc = 48.44657990268

Inradius: r = 2.1843950083
Circumradius: R = 24.84766283298

Vertex coordinates: A[4.69223583223; 0] B[0; 0] C[12.682213321; 47.33303654882]
Centroid: CG[5.79114971774; 15.77767884961]
Coordinates of the circumscribed circle: U[2.34661791611; 24.73656096085]
Coordinates of the inscribed circle: I[2.84661791611; 2.1843950083]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 80.41883100098° = 80°25'6″ = 1.7388028365 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 174.582168999° = 174°34'54″ = 0.09545673496 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 49 ; ; b = 48 ; ; beta = 75° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 48**2 = 49**2 + c**2 -2 * 49 * c * cos (75° ) ; ; ; ; c**2 -25.364c +97 =0 ; ; p=1; q=-25.364; r=97 ; ; D = q**2 - 4pr = 25.364**2 - 4 * 1 * 97 = 255.346011027 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 25.36 ± sqrt{ 255.35 } }{ 2 } ; ; c_{1,2} = 12.68213321 ± 7.98977488774 ; ; c_{1} = 20.6719080977 ; ; : Nr. 1
c_{2} = 4.69235832226 ; ; ; ; text{ Factored form: } ; ; (c -20.6719080977) (c -4.69235832226) = 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 = 49 ; ; b = 48 ; ; c = 4.69 ; ;

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

p = a+b+c = 49+48+4.69 = 101.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 101.69 }{ 2 } = 50.85 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 50.85 * (50.85-49)(50.85-48)(50.85-4.69) } ; ; T = sqrt{ 12331.11 } = 111.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 111.05 }{ 49 } = 4.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 111.05 }{ 48 } = 4.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 111.05 }{ 4.69 } = 47.33 ; ;

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{ 48**2+4.69**2-49**2 }{ 2 * 48 * 4.69 } ) = 99° 34'54" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 49**2+4.69**2-48**2 }{ 2 * 49 * 4.69 } ) = 75° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 49**2+48**2-4.69**2 }{ 2 * 49 * 48 } ) = 5° 25'6" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 111.05 }{ 50.85 } = 2.18 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 49 }{ 2 * sin 99° 34'54" } = 24.85 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 4.69**2 - 49**2 } }{ 2 } = 23.723 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.69**2+2 * 49**2 - 48**2 } }{ 2 } = 25.209 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 49**2 - 4.69**2 } }{ 2 } = 48.446 ; ;
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