Triangle calculator SSA

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Triangle has two solutions with side c=2.88880901303 and with side c=1.66219979929

#1 Acute scalene triangle.

Sides: a = 3.4   b = 2.6   c = 2.88880901303

Area: T = 3.64986577009
Perimeter: p = 8.88880901303
Semiperimeter: s = 4.44440450651

Angle ∠ A = α = 76.36219847179° = 76°21'43″ = 1.33327680567 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 55.63880152821° = 55°38'17″ = 0.97110665559 rad

Height: ha = 2.14662692358
Height: hb = 2.80766597699
Height: hc = 2.52766924066

Median: ma = 2.15988266027
Median: mb = 2.87441141766
Median: mc = 2.66598371848

Inradius: r = 0.821102176
Circumradius: R = 1.74993225485

Vertex coordinates: A[2.88880901303; 0] B[0; 0] C[2.27550440616; 2.52766924066]
Centroid: CG[1.72110447306; 0.84222308022]
Coordinates of the circumscribed circle: U[1.44440450651; 0.98773516235]
Coordinates of the inscribed circle: I[1.84440450651; 0.821102176]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.6388015282° = 103°38'17″ = 1.33327680567 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 124.3621984718° = 124°21'43″ = 0.97110665559 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 3.4 ; ; b = 2.6 ; ; beta = 48° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.6**2 = 3.4**2 + c**2 -2 * 3.4 * c * cos (48° ) ; ; ; ; c**2 -4.55c +4.8 =0 ; ; p=1; q=-4.55; r=4.8 ; ; D = q**2 - 4pr = 4.55**2 - 4 * 1 * 4.8 = 1.50330192925 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.55 ± sqrt{ 1.5 } }{ 2 } ; ; c_{1,2} = 2.27504406 ± 0.613046068671 ; ; c_{1} = 2.88809012867 ; ;
c_{2} = 1.66199799133 ; ; ; ; text{ Factored form: } ; ; (c -2.88809012867) (c -1.66199799133) = 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 = 3.4 ; ; b = 2.6 ; ; c = 2.89 ; ;

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

p = a+b+c = 3.4+2.6+2.89 = 8.89 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8.89 }{ 2 } = 4.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.44 * (4.44-3.4)(4.44-2.6)(4.44-2.89) } ; ; T = sqrt{ 13.31 } = 3.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.65 }{ 3.4 } = 2.15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.65 }{ 2.6 } = 2.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.65 }{ 2.89 } = 2.53 ; ;

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{ 2.6**2+2.89**2-3.4**2 }{ 2 * 2.6 * 2.89 } ) = 76° 21'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.4**2+2.89**2-2.6**2 }{ 2 * 3.4 * 2.89 } ) = 48° ; ; gamma = 180° - alpha - beta = 180° - 76° 21'43" - 48° = 55° 38'17" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.65 }{ 4.44 } = 0.82 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.4 }{ 2 * sin 76° 21'43" } = 1.75 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.6**2+2 * 2.89**2 - 3.4**2 } }{ 2 } = 2.159 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.89**2+2 * 3.4**2 - 2.6**2 } }{ 2 } = 2.874 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.6**2+2 * 3.4**2 - 2.89**2 } }{ 2 } = 2.66 ; ;







#2 Obtuse scalene triangle.

Sides: a = 3.4   b = 2.6   c = 1.66219979929

Area: T = 2.10996788543
Perimeter: p = 7.66219979929
Semiperimeter: s = 3.83109989965

Angle ∠ A = α = 103.6388015282° = 103°38'17″ = 1.80988245969 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 28.36219847179° = 28°21'43″ = 0.49550100157 rad

Height: ha = 1.23551052084
Height: hb = 1.61551375802
Height: hc = 2.52766924066

Median: ma = 1.36878883961
Median: mb = 2.33990422536
Median: mc = 2.91102303462

Inradius: r = 0.54880760648
Circumradius: R = 1.74993225485

Vertex coordinates: A[1.66219979929; 0] B[0; 0] C[2.27550440616; 2.52766924066]
Centroid: CG[1.31223473515; 0.84222308022]
Coordinates of the circumscribed circle: U[0.83109989965; 1.53993407831]
Coordinates of the inscribed circle: I[1.23109989965; 0.54880760648]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 76.36219847179° = 76°21'43″ = 1.80988245969 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 151.6388015282° = 151°38'17″ = 0.49550100157 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 3.4 ; ; b = 2.6 ; ; beta = 48° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.6**2 = 3.4**2 + c**2 -2 * 3.4 * c * cos (48° ) ; ; ; ; c**2 -4.55c +4.8 =0 ; ; p=1; q=-4.55; r=4.8 ; ; D = q**2 - 4pr = 4.55**2 - 4 * 1 * 4.8 = 1.50330192925 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.55 ± sqrt{ 1.5 } }{ 2 } ; ; c_{1,2} = 2.27504406 ± 0.613046068671 ; ; c_{1} = 2.88809012867 ; ; : Nr. 1
c_{2} = 1.66199799133 ; ; ; ; text{ Factored form: } ; ; (c -2.88809012867) (c -1.66199799133) = 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 = 3.4 ; ; b = 2.6 ; ; c = 1.66 ; ;

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

p = a+b+c = 3.4+2.6+1.66 = 7.66 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.66 }{ 2 } = 3.83 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.83 * (3.83-3.4)(3.83-2.6)(3.83-1.66) } ; ; T = sqrt{ 4.41 } = 2.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.1 }{ 3.4 } = 1.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.1 }{ 2.6 } = 1.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.1 }{ 1.66 } = 2.53 ; ;

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{ 2.6**2+1.66**2-3.4**2 }{ 2 * 2.6 * 1.66 } ) = 103° 38'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.4**2+1.66**2-2.6**2 }{ 2 * 3.4 * 1.66 } ) = 48° ; ; gamma = 180° - alpha - beta = 180° - 103° 38'17" - 48° = 28° 21'43" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.1 }{ 3.83 } = 0.55 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.4 }{ 2 * sin 103° 38'17" } = 1.75 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.6**2+2 * 1.66**2 - 3.4**2 } }{ 2 } = 1.368 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.66**2+2 * 3.4**2 - 2.6**2 } }{ 2 } = 2.339 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.6**2+2 * 3.4**2 - 1.66**2 } }{ 2 } = 2.91 ; ;
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