Triangle calculator SSA

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Triangle has two solutions with side c=5387.486622832 and with side c=2579.589896061

#1 Obtuse scalene triangle.

Sides: a = 4000   b = 1450   c = 5387.486622832

Area: T = 976563.5555305
Perimeter: p = 10837.48662283
Semiperimeter: s = 5418.743311416

Angle ∠ A = α = 14.47987495877° = 14°28'44″ = 0.25327018519 rad
Angle ∠ B = β = 5.2° = 5°12' = 0.09107571211 rad
Angle ∠ C = γ = 160.3211250412° = 160°19'17″ = 2.79881336806 rad

Height: ha = 488.2821777653
Height: hb = 1346.984421421
Height: hc = 362.5330320791

Median: ma = 3400.552200374
Median: mb = 4689.017684047
Median: mc = 1339.775536733

Inradius: r = 180.2219570246
Circumradius: R = 7999.331090747

Vertex coordinates: A[5387.486622832; 0] B[0; 0] C[3983.538759446; 362.5330320791]
Centroid: CG[3123.675460759; 120.8433440264]
Coordinates of the circumscribed circle: U[2693.743311416; -7532.13440271]
Coordinates of the inscribed circle: I[3968.743311416; 180.2219570246]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.5211250412° = 165°31'17″ = 0.25327018519 rad
∠ B' = β' = 174.8° = 174°48' = 0.09107571211 rad
∠ C' = γ' = 19.67987495877° = 19°40'43″ = 2.79881336806 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 4000 ; ; b = 1450 ; ; beta = 5° 12' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 1450**2 = 4000**2 + c**2 -2 * 4000 * c * cos (5° 12') ; ; ; ; c**2 -7967.075c +13897500 =0 ; ; p=1; q=-7967.075; r=13897500 ; ; D = q**2 - 4pr = 7967.075**2 - 4 * 1 * 13897500 = 7884287.06603 ; ;
D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7967.08 ± sqrt{ 7884287.07 } }{ 2 } ; ; c_{1,2} = 3983.53759446 ± 1403.94863386 ; ; c_{1} = 5387.48622832 ; ; c_{2} = 2579.58896061 ; ; ; ; text{ Factored form: } ; ; (c -5387.48622832) (c -2579.58896061) = 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 = 4000 ; ; b = 1450 ; ; c = 5387.49 ; ;

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

p = a+b+c = 4000+1450+5387.49 = 10837.49 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10837.49 }{ 2 } = 5418.74 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5418.74 * (5418.74-4000)(5418.74-1450)(5418.74-5387.49) } ; ; T = sqrt{ 953676377551 } = 976563.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 976563.56 }{ 4000 } = 488.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 976563.56 }{ 1450 } = 1346.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 976563.56 }{ 5387.49 } = 362.53 ; ;

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{ 1450**2+5387.49**2-4000**2 }{ 2 * 1450 * 5387.49 } ) = 14° 28'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4000**2+5387.49**2-1450**2 }{ 2 * 4000 * 5387.49 } ) = 5° 12' ; ;
gamma = 180° - alpha - beta = 180° - 14° 28'44" - 5° 12' = 160° 19'17" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 976563.56 }{ 5418.74 } = 180.22 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4000 }{ 2 * sin 14° 28'44" } = 7999.33 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1450**2+2 * 5387.49**2 - 4000**2 } }{ 2 } = 3400.552 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5387.49**2+2 * 4000**2 - 1450**2 } }{ 2 } = 4689.017 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1450**2+2 * 4000**2 - 5387.49**2 } }{ 2 } = 1339.775 ; ;



#2 Obtuse scalene triangle.

Sides: a = 4000   b = 1450   c = 2579.589896061

Area: T = 467589.6076699
Perimeter: p = 8029.589896061
Semiperimeter: s = 4014.79444803

Angle ∠ A = α = 165.5211250412° = 165°31'17″ = 2.88988908017 rad
Angle ∠ B = β = 5.2° = 5°12' = 0.09107571211 rad
Angle ∠ C = γ = 9.27987495877° = 9°16'44″ = 0.16219447308 rad

Height: ha = 233.795480335
Height: hb = 644.9511181654
Height: hc = 362.5330320791

Median: ma = 615.1343808894
Median: mb = 3286.566577644
Median: mc = 2718.029873395

Inradius: r = 116.4676635837
Circumradius: R = 7999.331090747

Vertex coordinates: A[2579.589896061; 0] B[0; 0] C[3983.538759446; 362.5330320791]
Centroid: CG[2187.709885169; 120.8433440264]
Coordinates of the circumscribed circle: U[1289.79444803; 7894.664434789]
Coordinates of the inscribed circle: I[2564.79444803; 116.4676635837]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 14.47987495877° = 14°28'44″ = 2.88988908017 rad
∠ B' = β' = 174.8° = 174°48' = 0.09107571211 rad
∠ C' = γ' = 170.7211250412° = 170°43'17″ = 0.16219447308 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 4000 ; ; b = 1450 ; ; beta = 5° 12' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 1450**2 = 4000**2 + c**2 -2 * 4000 * c * cos (5° 12') ; ; ; ; c**2 -7967.075c +13897500 =0 ; ; p=1; q=-7967.075; r=13897500 ; ; D = q**2 - 4pr = 7967.075**2 - 4 * 1 * 13897500 = 7884287.06603 ; ; : Nr. 1
D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7967.08 ± sqrt{ 7884287.07 } }{ 2 } ; ; c_{1,2} = 3983.53759446 ± 1403.94863386 ; ; c_{1} = 5387.48622832 ; ; c_{2} = 2579.58896061 ; ; ; ; text{ Factored form: } ; ; (c -5387.48622832) (c -2579.58896061) = 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 = 4000 ; ; b = 1450 ; ; c = 2579.59 ; ;

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

p = a+b+c = 4000+1450+2579.59 = 8029.59 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8029.59 }{ 2 } = 4014.79 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4014.79 * (4014.79-4000)(4014.79-1450)(4014.79-2579.59) } ; ; T = sqrt{ 218640040293 } = 467589.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 467589.61 }{ 4000 } = 233.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 467589.61 }{ 1450 } = 644.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 467589.61 }{ 2579.59 } = 362.53 ; ;

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{ 1450**2+2579.59**2-4000**2 }{ 2 * 1450 * 2579.59 } ) = 165° 31'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4000**2+2579.59**2-1450**2 }{ 2 * 4000 * 2579.59 } ) = 5° 12' ; ;
gamma = 180° - alpha - beta = 180° - 165° 31'17" - 5° 12' = 9° 16'44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 467589.61 }{ 4014.79 } = 116.47 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4000 }{ 2 * sin 165° 31'17" } = 7999.33 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1450**2+2 * 2579.59**2 - 4000**2 } }{ 2 } = 615.134 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2579.59**2+2 * 4000**2 - 1450**2 } }{ 2 } = 3286.566 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1450**2+2 * 4000**2 - 2579.59**2 } }{ 2 } = 2718.029 ; ;
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