Triangle calculator SSA

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Triangle has two solutions with side c=3.67767673279 and with side c=0.5587554998

#1 Obtuse scalene triangle.

Sides: a = 2.3   b = 1.8   c = 3.67767673279

Area: T = 1.65221215643
Perimeter: p = 7.77767673279
Semiperimeter: s = 3.8888383664

Angle ∠ A = α = 29.95215535207° = 29°57'6″ = 0.5232753225 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 127.0488446479° = 127°2'54″ = 2.21774137006 rad

Height: ha = 1.43766274472
Height: hb = 1.8365690627
Height: hc = 0.89986815955

Median: ma = 2.65664655074
Median: mb = 2.93216051903
Median: mc = 0.94109280015

Inradius: r = 0.42548864585
Circumradius: R = 2.30333741987

Vertex coordinates: A[3.67767673279; 0] B[0; 0] C[2.11771611629; 0.89986815955]
Centroid: CG[1.9311309497; 0.32995605318]
Coordinates of the circumscribed circle: U[1.8388383664; -1.3887760139]
Coordinates of the inscribed circle: I[2.0888383664; 0.42548864585]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0488446479° = 150°2'54″ = 0.5232753225 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 52.95215535207° = 52°57'6″ = 2.21774137006 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 2.3 ; ; b = 1.8 ; ; beta = 23° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 1.8**2 = 2.3**2 + c**2 -2 * 2.3 * c * cos (23° ) ; ; ; ; c**2 -4.234c +2.05 =0 ; ; p=1; q=-4.234; r=2.05 ; ; D = q**2 - 4pr = 4.234**2 - 4 * 1 * 2.05 = 9.72948555946 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.23 ± sqrt{ 9.73 } }{ 2 } ; ;
c_{1,2} = 2.11716116 ± 1.55960616499 ; ; c_{1} = 3.67676732793 ; ; c_{2} = 0.557554997954 ; ; ; ; text{ Factored form: } ; ; (c -3.67676732793) (c -0.557554997954) = 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 = 2.3 ; ; b = 1.8 ; ; c = 3.68 ; ;

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

p = a+b+c = 2.3+1.8+3.68 = 7.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.78 }{ 2 } = 3.89 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.89 * (3.89-2.3)(3.89-1.8)(3.89-3.68) } ; ; T = sqrt{ 2.73 } = 1.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.65 }{ 2.3 } = 1.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.65 }{ 1.8 } = 1.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.65 }{ 3.68 } = 0.9 ; ;

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{ 1.8**2+3.68**2-2.3**2 }{ 2 * 1.8 * 3.68 } ) = 29° 57'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.3**2+3.68**2-1.8**2 }{ 2 * 2.3 * 3.68 } ) = 23° ; ;
gamma = 180° - alpha - beta = 180° - 29° 57'6" - 23° = 127° 2'54" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.65 }{ 3.89 } = 0.42 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.3 }{ 2 * sin 29° 57'6" } = 2.3 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.8**2+2 * 3.68**2 - 2.3**2 } }{ 2 } = 2.656 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.68**2+2 * 2.3**2 - 1.8**2 } }{ 2 } = 2.932 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.8**2+2 * 2.3**2 - 3.68**2 } }{ 2 } = 0.941 ; ;



#2 Obtuse scalene triangle.

Sides: a = 2.3   b = 1.8   c = 0.5587554998

Area: T = 0.25105322076
Perimeter: p = 4.6587554998
Semiperimeter: s = 2.3298777499

Angle ∠ A = α = 150.0488446479° = 150°2'54″ = 2.61988394286 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 6.95215535207° = 6°57'6″ = 0.12113274971 rad

Height: ha = 0.21878540935
Height: hb = 0.27883691195
Height: hc = 0.89986815955

Median: ma = 0.67330035571
Median: mb = 1.41108273416
Median: mc = 2.04662851967

Inradius: r = 0.10875809981
Circumradius: R = 2.30333741987

Vertex coordinates: A[0.5587554998; 0] B[0; 0] C[2.11771611629; 0.89986815955]
Centroid: CG[0.89215720536; 0.32995605318]
Coordinates of the circumscribed circle: U[0.2798777499; 2.28664417345]
Coordinates of the inscribed circle: I[0.5298777499; 0.10875809981]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.95215535207° = 29°57'6″ = 2.61988394286 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 173.0488446479° = 173°2'54″ = 0.12113274971 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 2.3 ; ; b = 1.8 ; ; beta = 23° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 1.8**2 = 2.3**2 + c**2 -2 * 2.3 * c * cos (23° ) ; ; ; ; c**2 -4.234c +2.05 =0 ; ; p=1; q=-4.234; r=2.05 ; ; D = q**2 - 4pr = 4.234**2 - 4 * 1 * 2.05 = 9.72948555946 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.23 ± sqrt{ 9.73 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 2.11716116 ± 1.55960616499 ; ; c_{1} = 3.67676732793 ; ; c_{2} = 0.557554997954 ; ; ; ; text{ Factored form: } ; ; (c -3.67676732793) (c -0.557554997954) = 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 = 2.3 ; ; b = 1.8 ; ; c = 0.56 ; ;

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

p = a+b+c = 2.3+1.8+0.56 = 4.66 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.66 }{ 2 } = 2.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.33 * (2.33-2.3)(2.33-1.8)(2.33-0.56) } ; ; T = sqrt{ 0.06 } = 0.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.25 }{ 2.3 } = 0.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.25 }{ 1.8 } = 0.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.25 }{ 0.56 } = 0.9 ; ;

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{ 1.8**2+0.56**2-2.3**2 }{ 2 * 1.8 * 0.56 } ) = 150° 2'54" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.3**2+0.56**2-1.8**2 }{ 2 * 2.3 * 0.56 } ) = 23° ; ;
gamma = 180° - alpha - beta = 180° - 150° 2'54" - 23° = 6° 57'6" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.25 }{ 2.33 } = 0.11 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.3 }{ 2 * sin 150° 2'54" } = 2.3 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.8**2+2 * 0.56**2 - 2.3**2 } }{ 2 } = 0.673 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.56**2+2 * 2.3**2 - 1.8**2 } }{ 2 } = 1.411 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.8**2+2 * 2.3**2 - 0.56**2 } }{ 2 } = 2.046 ; ;
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