Triangle calculator SSA

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Triangle has two solutions with side c=3.92200877706 and with side c=0.12549972013

#1 Obtuse scalene triangle.

Sides: a = 2.5   b = 2.4   c = 3.92200877706

Area: T = 2.88802122241
Perimeter: p = 8.82200877706
Semiperimeter: s = 4.41100438853

Angle ∠ A = α = 37.75442772144° = 37°45'15″ = 0.65989364441 rad
Angle ∠ B = β = 36° = 0.62883185307 rad
Angle ∠ C = γ = 106.2465722786° = 106°14'45″ = 1.85443376788 rad

Height: ha = 2.30441697793
Height: hb = 2.44001768534
Height: hc = 1.46994631307

Median: ma = 33.0001740057
Median: mb = 3.06108077471
Median: mc = 1.47107916126

Inradius: r = 0.65331028486
Circumradius: R = 2.042156194

Vertex coordinates: A[3.92200877706; 0] B[0; 0] C[2.02325424859; 1.46994631307]
Centroid: CG[1.98108767522; 0.49898210436]
Coordinates of the circumscribed circle: U[1.96600438853; -0.57111419462]
Coordinates of the inscribed circle: I[2.01100438853; 0.65331028486]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.2465722786° = 142°14'45″ = 0.65989364441 rad
∠ B' = β' = 144° = 0.62883185307 rad
∠ C' = γ' = 73.75442772144° = 73°45'15″ = 1.85443376788 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 2.5 ; ; b = 2.4 ; ; beta = 36° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.4**2 = 2.5**2 + c**2 -2 * 2.5 * c * cos (36° ) ; ; ; ; c**2 -4.045c +0.49 =0 ; ; p=1; q=-4.045; r=0.49 ; ; D = q**2 - 4pr = 4.045**2 - 4 * 1 * 0.49 = 14.4027124297 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.05 ± sqrt{ 14.4 } }{ 2 } ; ; c_{1,2} = 2.02254249 ± 1.89754528468 ; ; c_{1} = 3.92008777468 ; ; c_{2} = 0.124997205317 ; ; ; ; text{ Factored form: } ; ; (c -3.92008777468) (c -0.124997205317) = 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 = 2.5 ; ; b = 2.4 ; ; c = 3.92 ; ;

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

p = a+b+c = 2.5+2.4+3.92 = 8.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8.82 }{ 2 } = 4.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.41 * (4.41-2.5)(4.41-2.4)(4.41-3.92) } ; ; T = sqrt{ 8.3 } = 2.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.88 }{ 2.5 } = 2.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.88 }{ 2.4 } = 2.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.88 }{ 3.92 } = 1.47 ; ;

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{ 2.4**2+3.92**2-2.5**2 }{ 2 * 2.4 * 3.92 } ) = 37° 45'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.5**2+3.92**2-2.4**2 }{ 2 * 2.5 * 3.92 } ) = 36° ; ; gamma = 180° - alpha - beta = 180° - 37° 45'15" - 36° = 106° 14'45" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.88 }{ 4.41 } = 0.65 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.5 }{ 2 * sin 37° 45'15" } = 2.04 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 3.92**2 - 2.5**2 } }{ 2 } = 3 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.92**2+2 * 2.5**2 - 2.4**2 } }{ 2 } = 3.061 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 2.5**2 - 3.92**2 } }{ 2 } = 1.471 ; ;







#2 Obtuse scalene triangle.

Sides: a = 2.5   b = 2.4   c = 0.12549972013

Area: T = 0.09218393893
Perimeter: p = 5.02549972013
Semiperimeter: s = 2.51224986006

Angle ∠ A = α = 142.2465722786° = 142°14'45″ = 2.48326562095 rad
Angle ∠ B = β = 36° = 0.62883185307 rad
Angle ∠ C = γ = 1.75442772144° = 1°45'15″ = 0.03106179134 rad

Height: ha = 0.07334715115
Height: hb = 0.07765328245
Height: hc = 1.46994631307

Median: ma = 1.15112220247
Median: mb = 1.30110811466
Median: mc = 2.45497130291

Inradius: r = 0.03765530111
Circumradius: R = 2.042156194

Vertex coordinates: A[0.12549972013; 0] B[0; 0] C[2.02325424859; 1.46994631307]
Centroid: CG[0.71658465624; 0.49898210436]
Coordinates of the circumscribed circle: U[0.06224986006; 2.04106050769]
Coordinates of the inscribed circle: I[0.11224986006; 0.03765530111]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 37.75442772144° = 37°45'15″ = 2.48326562095 rad
∠ B' = β' = 144° = 0.62883185307 rad
∠ C' = γ' = 178.2465722786° = 178°14'45″ = 0.03106179134 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 2.5 ; ; b = 2.4 ; ; beta = 36° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.4**2 = 2.5**2 + c**2 -2 * 2.5 * c * cos (36° ) ; ; ; ; c**2 -4.045c +0.49 =0 ; ; p=1; q=-4.045; r=0.49 ; ; D = q**2 - 4pr = 4.045**2 - 4 * 1 * 0.49 = 14.4027124297 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 4.05 ± sqrt{ 14.4 } }{ 2 } ; ; c_{1,2} = 2.02254249 ± 1.89754528468 ; ; c_{1} = 3.92008777468 ; ; c_{2} = 0.124997205317 ; ; ; ; text{ Factored form: } ; ; (c -3.92008777468) (c -0.124997205317) = 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 = 2.5 ; ; b = 2.4 ; ; c = 0.12 ; ;

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

p = a+b+c = 2.5+2.4+0.12 = 5.02 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.02 }{ 2 } = 2.51 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.51 * (2.51-2.5)(2.51-2.4)(2.51-0.12) } ; ; T = sqrt{ 0.01 } = 0.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.09 }{ 2.5 } = 0.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.09 }{ 2.4 } = 0.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.09 }{ 0.12 } = 1.47 ; ;

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{ 2.4**2+0.12**2-2.5**2 }{ 2 * 2.4 * 0.12 } ) = 142° 14'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.5**2+0.12**2-2.4**2 }{ 2 * 2.5 * 0.12 } ) = 36° ; ; gamma = 180° - alpha - beta = 180° - 142° 14'45" - 36° = 1° 45'15" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.09 }{ 2.51 } = 0.04 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.5 }{ 2 * sin 142° 14'45" } = 2.04 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 0.12**2 - 2.5**2 } }{ 2 } = 1.151 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.12**2+2 * 2.5**2 - 2.4**2 } }{ 2 } = 1.301 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.4**2+2 * 2.5**2 - 0.12**2 } }{ 2 } = 2.45 ; ;
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