Triangle calculator SSA

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Triangle has two solutions with side c=7.42552681386 and with side c=2.17550056292

#1 Obtuse scalene triangle.

Sides: a = 5.6   b = 3.9   c = 7.42552681386

Area: T = 10.70880282619
Perimeter: p = 16.92552681386
Semiperimeter: s = 8.46326340693

Angle ∠ A = α = 47.69224029317° = 47°41'33″ = 0.83223894593 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 101.3087597068° = 101°18'27″ = 1.76881511261 rad

Height: ha = 3.82442958078
Height: hb = 5.49112965446
Height: hc = 2.88442132195

Median: ma = 5.22880305532
Median: mb = 6.28105098094
Median: mc = 3.08224257116

Inradius: r = 1.2655330413
Circumradius: R = 3.78661278515

Vertex coordinates: A[7.42552681386; 0] B[0; 0] C[4.88001368839; 2.88442132195]
Centroid: CG[4.07551350075; 0.96114044065]
Coordinates of the circumscribed circle: U[3.71326340693; -0.74223694331]
Coordinates of the inscribed circle: I[4.56326340693; 1.2655330413]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.3087597068° = 132°18'27″ = 0.83223894593 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 78.69224029317° = 78°41'33″ = 1.76881511261 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 5.6 ; ; b = 3.9 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 3.9**2 = 5.6**2 + c**2 -2 * 5.6 * c * cos (31° ) ; ; ; ; c**2 -9.6c +16.15 =0 ; ; p=1; q=-9.6; r=16.15 ; ; D = q**2 - 4pr = 9.6**2 - 4 * 1 * 16.15 = 27.5652564179 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9.6 ± sqrt{ 27.57 } }{ 2 } ; ; c_{1,2} = 4.80013688 ± 2.62513125472 ; ; c_{1} = 7.42526813472 ; ;
c_{2} = 2.17500562528 ; ; ; ; text{ Factored form: } ; ; (c -7.42526813472) (c -2.17500562528) = 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 = 5.6 ; ; b = 3.9 ; ; c = 7.43 ; ;

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

p = a+b+c = 5.6+3.9+7.43 = 16.93 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.93 }{ 2 } = 8.46 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.46 * (8.46-5.6)(8.46-3.9)(8.46-7.43) } ; ; T = sqrt{ 114.66 } = 10.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.71 }{ 5.6 } = 3.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.71 }{ 3.9 } = 5.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.71 }{ 7.43 } = 2.88 ; ;

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{ 3.9**2+7.43**2-5.6**2 }{ 2 * 3.9 * 7.43 } ) = 47° 41'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.6**2+7.43**2-3.9**2 }{ 2 * 5.6 * 7.43 } ) = 31° ; ; gamma = 180° - alpha - beta = 180° - 47° 41'33" - 31° = 101° 18'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.71 }{ 8.46 } = 1.27 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.6 }{ 2 * sin 47° 41'33" } = 3.79 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 7.43**2 - 5.6**2 } }{ 2 } = 5.228 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.43**2+2 * 5.6**2 - 3.9**2 } }{ 2 } = 6.281 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 5.6**2 - 7.43**2 } }{ 2 } = 3.082 ; ;







#2 Obtuse scalene triangle.

Sides: a = 5.6   b = 3.9   c = 2.17550056292

Area: T = 3.13765899941
Perimeter: p = 11.67550056292
Semiperimeter: s = 5.83875028146

Angle ∠ A = α = 132.3087597068° = 132°18'27″ = 2.30992031942 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 16.69224029317° = 16°41'33″ = 0.29113373912 rad

Height: ha = 1.12202107122
Height: hb = 1.60985076893
Height: hc = 2.88442132195

Median: ma = 1.4659563203
Median: mb = 3.7743966712
Median: mc = 4.70113123304

Inradius: r = 0.53773170847
Circumradius: R = 3.78661278515

Vertex coordinates: A[2.17550056292; 0] B[0; 0] C[4.88001368839; 2.88442132195]
Centroid: CG[2.32550475044; 0.96114044065]
Coordinates of the circumscribed circle: U[1.08875028146; 3.62765826526]
Coordinates of the inscribed circle: I[1.93875028146; 0.53773170847]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 47.69224029317° = 47°41'33″ = 2.30992031942 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 163.3087597068° = 163°18'27″ = 0.29113373912 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 5.6 ; ; b = 3.9 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 3.9**2 = 5.6**2 + c**2 -2 * 5.6 * c * cos (31° ) ; ; ; ; c**2 -9.6c +16.15 =0 ; ; p=1; q=-9.6; r=16.15 ; ; D = q**2 - 4pr = 9.6**2 - 4 * 1 * 16.15 = 27.5652564179 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9.6 ± sqrt{ 27.57 } }{ 2 } ; ; c_{1,2} = 4.80013688 ± 2.62513125472 ; ; c_{1} = 7.42526813472 ; ; : Nr. 1
c_{2} = 2.17500562528 ; ; ; ; text{ Factored form: } ; ; (c -7.42526813472) (c -2.17500562528) = 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 = 5.6 ; ; b = 3.9 ; ; c = 2.18 ; ;

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

p = a+b+c = 5.6+3.9+2.18 = 11.68 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.68 }{ 2 } = 5.84 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.84 * (5.84-5.6)(5.84-3.9)(5.84-2.18) } ; ; T = sqrt{ 9.84 } = 3.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.14 }{ 5.6 } = 1.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.14 }{ 3.9 } = 1.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.14 }{ 2.18 } = 2.88 ; ;

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{ 3.9**2+2.18**2-5.6**2 }{ 2 * 3.9 * 2.18 } ) = 132° 18'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.6**2+2.18**2-3.9**2 }{ 2 * 5.6 * 2.18 } ) = 31° ; ; gamma = 180° - alpha - beta = 180° - 132° 18'27" - 31° = 16° 41'33" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.14 }{ 5.84 } = 0.54 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.6 }{ 2 * sin 132° 18'27" } = 3.79 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 2.18**2 - 5.6**2 } }{ 2 } = 1.46 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.18**2+2 * 5.6**2 - 3.9**2 } }{ 2 } = 3.774 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.9**2+2 * 5.6**2 - 2.18**2 } }{ 2 } = 4.701 ; ;
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