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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 48**2 = 49**2 + c**2 -2 * 48 * c * cos (75° ) ; ; ; ; c**2 -25.364c +97 =0 ; ; p=1; q=-25.36426642; 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} = 20.6719080978 ; ;
c_{2} = 4.69235832228 ; ; ; ; (c -20.6719080978) (c -4.69235832228) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 49**2-48**2-4.69**2 }{ 2 * 48 * 4.69 } ) = 99° 34'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 48**2-49**2-4.69**2 }{ 2 * 49 * 4.69 } ) = 75° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.69**2-49**2-48**2 }{ 2 * 48 * 49 } ) = 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 ; ;




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