Triangle calculator SSA

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Triangle has two solutions with side c=104.3221603047 and with side c=48.8877285577

#1 Acute scalene triangle.

Sides: a = 100   b = 70   c = 104.3221603047

Area: T = 3352.832169306
Perimeter: p = 274.3221603047
Semiperimeter: s = 137.1610801523

Angle ∠ A = α = 66.67441765214° = 66°40'27″ = 1.16436839064 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 73.32658234786° = 73°19'33″ = 1.28797770464 rad

Height: ha = 67.05766338611
Height: hb = 95.79551912302
Height: hc = 64.27987609687

Median: ma = 73.42768236486
Median: mb = 96.00325959603
Median: mc = 68.77695483804

Inradius: r = 24.4454532664
Circumradius: R = 54.45503339401

Vertex coordinates: A[104.3221603047; 0] B[0; 0] C[76.60444443119; 64.27987609687]
Centroid: CG[60.30986824529; 21.42662536562]
Coordinates of the circumscribed circle: U[52.16108015234; 15.62333687349]
Coordinates of the inscribed circle: I[67.16108015234; 24.4454532664]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 113.3265823479° = 113°19'33″ = 1.16436839064 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 106.6744176521° = 106°40'27″ = 1.28797770464 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 = 100 ; ; b = 70 ; ; c = 104.32 ; ;

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

p = a+b+c = 100+70+104.32 = 274.32 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 274.32 }{ 2 } = 137.16 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 137.16 * (137.16-100)(137.16-70)(137.16-104.32) } ; ; T = sqrt{ 11241480.36 } = 3352.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3352.83 }{ 100 } = 67.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3352.83 }{ 70 } = 95.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3352.83 }{ 104.32 } = 64.28 ; ;

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{ 100**2-70**2-104.32**2 }{ 2 * 70 * 104.32 } ) = 66° 40'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 70**2-100**2-104.32**2 }{ 2 * 100 * 104.32 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 104.32**2-100**2-70**2 }{ 2 * 70 * 100 } ) = 73° 19'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3352.83 }{ 137.16 } = 24.44 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 66° 40'27" } = 54.45 ; ;





#2 Obtuse scalene triangle.

Sides: a = 100   b = 70   c = 48.8877285577

Area: T = 1571.207707201
Perimeter: p = 218.8877285577
Semiperimeter: s = 109.4443642789

Angle ∠ A = α = 113.3265823479° = 113°19'33″ = 1.97879087472 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 26.67441765214° = 26°40'27″ = 0.46655522056 rad

Height: ha = 31.42441414401
Height: hb = 44.89216306287
Height: hc = 64.27987609687

Median: ma = 33.83876025384
Median: mb = 70.49881088083
Median: mc = 82.78798787582

Inradius: r = 14.3566311906
Circumradius: R = 54.45503339401

Vertex coordinates: A[48.8877285577; 0] B[0; 0] C[76.60444443119; 64.27987609687]
Centroid: CG[41.83105766296; 21.42662536562]
Coordinates of the circumscribed circle: U[24.44436427885; 48.65553922337]
Coordinates of the inscribed circle: I[39.44436427885; 14.3566311906]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 66.67441765214° = 66°40'27″ = 1.97879087472 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 153.3265823479° = 153°19'33″ = 0.46655522056 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 70 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 70**2 = 100**2 + c**2 -2 * 70 * c * cos (40° ) ; ; ; ; c**2 -153.209c +5100 =0 ; ; p=1; q=-153.208888624; r=5100 ; ; D = q**2 - 4pr = 153.209**2 - 4 * 1 * 5100 = 3072.96355334 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 153.21 ± sqrt{ 3072.96 } }{ 2 } ; ; c_{1,2} = 76.6044443119 ± 27.7171587349 ; ; c_{1} = 104.321603047 ; ;
c_{2} = 48.887285577 ; ; ; ; (c -104.321603047) (c -48.887285577) = 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 = 100 ; ; b = 70 ; ; c = 48.89 ; ;

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

p = a+b+c = 100+70+48.89 = 218.89 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 218.89 }{ 2 } = 109.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 109.44 * (109.44-100)(109.44-70)(109.44-48.89) } ; ; T = sqrt{ 2468691.66 } = 1571.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1571.21 }{ 100 } = 31.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1571.21 }{ 70 } = 44.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1571.21 }{ 48.89 } = 64.28 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 100**2-70**2-48.89**2 }{ 2 * 70 * 48.89 } ) = 113° 19'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 70**2-100**2-48.89**2 }{ 2 * 100 * 48.89 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48.89**2-100**2-70**2 }{ 2 * 70 * 100 } ) = 26° 40'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1571.21 }{ 109.44 } = 14.36 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 113° 19'33" } = 54.45 ; ;




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