Triangle calculator SSA

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: h_a = 488.2821777653
Height: h_b = 1346.984421421
Height: h_c = 362.5330320791

Median: m_a = 3400.552200374
Median: m_b = 4689.017684047
Median: m_c = 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?

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.

$a = 4000 ; ; b = 1450 ; ; c = 5387.49 ; ;$

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

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

2. Semiperimeter of the triangle

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

3. The triangle area using Heron's formula

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

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

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

5. Calculation of the inner angles of the triangle using a Law of Cosines

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

6. Inradius

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

7. Circumradius

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

#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: h_a = 233.795480335
Height: h_b = 644.9511181654
Height: h_c = 362.5330320791

Median: m_a = 615.1343808894
Median: m_b = 3286.566577644
Median: m_c = 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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

$a = 4000 ; ; b = 1450 ; ; beta = 5° 12' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 1450**2 = 4000**2 + c**2 -2 * 1450 * c * cos (5° 12') ; ; ; ; c**2 -7967.075c +13897500 =0 ; ; p=1; q=-7967.07518893; r=13897500 ; ; D = q**2 - 4pr = 7967.075**2 - 4 * 1 * 13897500 = 7884287.06603 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 7967.08 ± sqrt{ 7884287.07 } }{ 2 } ; ; c_{1,2} = 3983.53759446 ± 1403.94863386 ; ;$
$c_{1} = 5387.48622832 ; ; c_{2} = 2579.58896061 ; ; ; ; (c -5387.48622832) (c -2579.58896061) = 0 ; ; ; ; c>0 ; ;$

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.

$a = 4000 ; ; b = 1450 ; ; c = 2579.59 ; ;$

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

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

7. Inradius

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

8. Circumradius

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

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