Triangle calculator SSA

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Triangle has two solutions with side c=7.78797870838 and with side c=1.43106304119

#1 Acute scalene triangle.

Sides: a = 8.3   b = 7.6   c = 7.78797870838

Area: T = 26.86105676245
Perimeter: p = 23.68797870838
Semiperimeter: s = 11.84398935419

Angle ∠ A = α = 65.31100521554° = 65°18'36″ = 1.14398754448 rad
Angle ∠ B = β = 56.3° = 56°18' = 0.98326203689 rad
Angle ∠ C = γ = 58.39899478446° = 58°23'24″ = 1.019909684 rad

Height: ha = 6.47224259336
Height: hb = 7.06985704275
Height: hc = 6.90552192137

Median: ma = 6.47545689844
Median: mb = 7.09899607569
Median: mc = 6.94221702826

Inradius: r = 2.26986494207
Circumradius: R = 4.56875595552

Vertex coordinates: A[7.78797870838; 0] B[0; 0] C[4.60552087478; 6.90552192137]
Centroid: CG[4.12883319439; 2.30217397379]
Coordinates of the circumscribed circle: U[3.89898935419; 2.39440193238]
Coordinates of the inscribed circle: I[4.24398935419; 2.26986494207]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114.6989947845° = 114°41'24″ = 1.14398754448 rad
∠ B' = β' = 123.7° = 123°42' = 0.98326203689 rad
∠ C' = γ' = 121.6110052155° = 121°36'36″ = 1.019909684 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 8.3 ; ; b = 7.6 ; ; beta = 56° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.6**2 = 8.3**2 + c**2 -2 * 8.3 * c * cos (56° 18') ; ; ; ; c**2 -9.21c +11.13 =0 ; ; p=1; q=-9.21; r=11.13 ; ; D = q**2 - 4pr = 9.21**2 - 4 * 1 * 11.13 = 40.3117904439 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9.21 ± sqrt{ 40.31 } }{ 2 } ; ; c_{1,2} = 4.60520875 ± 3.17457833593 ; ; c_{1} = 7.77978708593 ; ; c_{2} = 1.43063041407 ; ; ; ; text{ Factored form: } ; ; (c -7.77978708593) (c -1.43063041407) = 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 = 8.3 ; ; b = 7.6 ; ; c = 7.78 ; ;

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

p = a+b+c = 8.3+7.6+7.78 = 23.68 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.68 }{ 2 } = 11.84 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.84 * (11.84-8.3)(11.84-7.6)(11.84-7.78) } ; ; T = sqrt{ 721.49 } = 26.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26.86 }{ 8.3 } = 6.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26.86 }{ 7.6 } = 7.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26.86 }{ 7.78 } = 6.91 ; ;

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{ 7.6**2+7.78**2-8.3**2 }{ 2 * 7.6 * 7.78 } ) = 65° 18'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+7.78**2-7.6**2 }{ 2 * 8.3 * 7.78 } ) = 56° 18' ; ; gamma = 180° - alpha - beta = 180° - 65° 18'36" - 56° 18' = 58° 23'24" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26.86 }{ 11.84 } = 2.27 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 65° 18'36" } = 4.57 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.6**2+2 * 7.78**2 - 8.3**2 } }{ 2 } = 6.475 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.78**2+2 * 8.3**2 - 7.6**2 } }{ 2 } = 7.09 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.6**2+2 * 8.3**2 - 7.78**2 } }{ 2 } = 6.942 ; ;







#2 Obtuse scalene triangle.

Sides: a = 8.3   b = 7.6   c = 1.43106304119

Area: T = 4.93994083039
Perimeter: p = 17.33106304119
Semiperimeter: s = 8.66553152059

Angle ∠ A = α = 114.6989947845° = 114°41'24″ = 2.00217172088 rad
Angle ∠ B = β = 56.3° = 56°18' = 0.98326203689 rad
Angle ∠ C = γ = 9.01100521554° = 9°36″ = 0.15772550759 rad

Height: ha = 1.19902188684
Height: hb = 1.32998442905
Height: hc = 6.90552192137

Median: ma = 3.56110183498
Median: mb = 4.58656680743
Median: mc = 7.92554857363

Inradius: r = 0.57700206151
Circumradius: R = 4.56875595552

Vertex coordinates: A[1.43106304119; 0] B[0; 0] C[4.60552087478; 6.90552192137]
Centroid: CG[2.01219463866; 2.30217397379]
Coordinates of the circumscribed circle: U[0.71553152059; 4.51111998898]
Coordinates of the inscribed circle: I[1.06553152059; 0.57700206151]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 65.31100521554° = 65°18'36″ = 2.00217172088 rad
∠ B' = β' = 123.7° = 123°42' = 0.98326203689 rad
∠ C' = γ' = 170.9989947845° = 170°59'24″ = 0.15772550759 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 8.3 ; ; b = 7.6 ; ; beta = 56° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7.6**2 = 8.3**2 + c**2 -2 * 8.3 * c * cos (56° 18') ; ; ; ; c**2 -9.21c +11.13 =0 ; ; p=1; q=-9.21; r=11.13 ; ; D = q**2 - 4pr = 9.21**2 - 4 * 1 * 11.13 = 40.3117904439 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9.21 ± sqrt{ 40.31 } }{ 2 } ; ; c_{1,2} = 4.60520875 ± 3.17457833593 ; ; c_{1} = 7.77978708593 ; ; c_{2} = 1.43063041407 ; ; ; ; text{ Factored form: } ; ; (c -7.77978708593) (c -1.43063041407) = 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 = 8.3 ; ; b = 7.6 ; ; c = 1.43 ; ;

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

p = a+b+c = 8.3+7.6+1.43 = 17.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.33 }{ 2 } = 8.67 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.67 * (8.67-8.3)(8.67-7.6)(8.67-1.43) } ; ; T = sqrt{ 24.4 } = 4.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.94 }{ 8.3 } = 1.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.94 }{ 7.6 } = 1.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.94 }{ 1.43 } = 6.91 ; ;

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{ 7.6**2+1.43**2-8.3**2 }{ 2 * 7.6 * 1.43 } ) = 114° 41'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+1.43**2-7.6**2 }{ 2 * 8.3 * 1.43 } ) = 56° 18' ; ; gamma = 180° - alpha - beta = 180° - 114° 41'24" - 56° 18' = 9° 36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.94 }{ 8.67 } = 0.57 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 114° 41'24" } = 4.57 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.6**2+2 * 1.43**2 - 8.3**2 } }{ 2 } = 3.561 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.43**2+2 * 8.3**2 - 7.6**2 } }{ 2 } = 4.586 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.6**2+2 * 8.3**2 - 1.43**2 } }{ 2 } = 7.925 ; ;
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