Triangle calculator SSA

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Triangle has two solutions with side c=322.8711376228 and with side c=23.53987852859

#1 Obtuse scalene triangle.

Sides: a = 200   b = 180   c = 322.8711376228

Area: T = 16143.56988114
Perimeter: p = 702.8711376228
Semiperimeter: s = 351.4365688114

Angle ∠ A = α = 33.74989885959° = 33°44'56″ = 0.58990309702 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 116.2511011404° = 116°15'4″ = 2.02989629078 rad

Height: ha = 161.4365688114
Height: hb = 179.3732986793
Height: hc = 100

Median: ma = 241.5011475759
Median: mb = 253.0287592949
Median: mc = 100.6990211059

Inradius: r = 45.9366054184
Circumradius: R = 180

Vertex coordinates: A[322.8711376228; 0] B[0; 0] C[173.2055080757; 100]
Centroid: CG[165.3598818995; 33.33333333333]
Coordinates of the circumscribed circle: U[161.4365688114; -79.61548139682]
Coordinates of the inscribed circle: I[171.4365688114; 45.9366054184]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.2511011404° = 146°15'4″ = 0.58990309702 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 63.74989885959° = 63°44'56″ = 2.02989629078 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 200 ; ; b = 180 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 180**2 = 200**2 + c**2 -2 * 200 * c * cos (30° ) ; ; ; ; c**2 -346.41c +7600 =0 ; ; p=1; q=-346.41; r=7600 ; ; D = q**2 - 4pr = 346.41**2 - 4 * 1 * 7600 = 89600 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 346.41 ± sqrt{ 89600 } }{ 2 } ; ; c_{1,2} = 173.20508076 ± 149.666295471 ; ; c_{1} = 322.871376231 ; ;
c_{2} = 23.538785289 ; ; ; ; text{ Factored form: } ; ; (c -322.871376231) (c -23.538785289) = 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 = 200 ; ; b = 180 ; ; c = 322.87 ; ;

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

p = a+b+c = 200+180+322.87 = 702.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 702.87 }{ 2 } = 351.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 351.44 * (351.44-200)(351.44-180)(351.44-322.87) } ; ; T = sqrt{ 260614813.97 } = 16143.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16143.57 }{ 200 } = 161.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16143.57 }{ 180 } = 179.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16143.57 }{ 322.87 } = 100 ; ;

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{ 180**2+322.87**2-200**2 }{ 2 * 180 * 322.87 } ) = 33° 44'56" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 200**2+322.87**2-180**2 }{ 2 * 200 * 322.87 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 33° 44'56" - 30° = 116° 15'4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16143.57 }{ 351.44 } = 45.94 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 200 }{ 2 * sin 33° 44'56" } = 180 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 180**2+2 * 322.87**2 - 200**2 } }{ 2 } = 241.501 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 322.87**2+2 * 200**2 - 180**2 } }{ 2 } = 253.028 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 180**2+2 * 200**2 - 322.87**2 } }{ 2 } = 100.69 ; ;







#2 Obtuse scalene triangle.

Sides: a = 200   b = 180   c = 23.53987852859

Area: T = 1176.93992643
Perimeter: p = 403.5398785286
Semiperimeter: s = 201.7699392643

Angle ∠ A = α = 146.2511011404° = 146°15'4″ = 2.55325616834 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 3.74989885959° = 3°44'56″ = 0.06554321946 rad

Height: ha = 11.7699392643
Height: hb = 13.07771029366
Height: hc = 100

Median: ma = 80.48800422861
Median: mb = 110.3549613531
Median: mc = 189.8998608201

Inradius: r = 5.83330911784
Circumradius: R = 180

Vertex coordinates: A[23.53987852859; 0] B[0; 0] C[173.2055080757; 100]
Centroid: CG[65.58112886809; 33.33333333333]
Coordinates of the circumscribed circle: U[11.7699392643; 179.6154813968]
Coordinates of the inscribed circle: I[21.7699392643; 5.83330911784]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 33.74989885959° = 33°44'56″ = 2.55325616834 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 176.2511011404° = 176°15'4″ = 0.06554321946 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 200 ; ; b = 180 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 180**2 = 200**2 + c**2 -2 * 200 * c * cos (30° ) ; ; ; ; c**2 -346.41c +7600 =0 ; ; p=1; q=-346.41; r=7600 ; ; D = q**2 - 4pr = 346.41**2 - 4 * 1 * 7600 = 89600 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 346.41 ± sqrt{ 89600 } }{ 2 } ; ; c_{1,2} = 173.20508076 ± 149.666295471 ; ; c_{1} = 322.871376231 ; ; : Nr. 1
c_{2} = 23.538785289 ; ; ; ; text{ Factored form: } ; ; (c -322.871376231) (c -23.538785289) = 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 = 200 ; ; b = 180 ; ; c = 23.54 ; ;

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

p = a+b+c = 200+180+23.54 = 403.54 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 403.54 }{ 2 } = 201.77 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 201.77 * (201.77-200)(201.77-180)(201.77-23.54) } ; ; T = sqrt{ 1385186.03 } = 1176.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1176.94 }{ 200 } = 11.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1176.94 }{ 180 } = 13.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1176.94 }{ 23.54 } = 100 ; ;

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{ 180**2+23.54**2-200**2 }{ 2 * 180 * 23.54 } ) = 146° 15'4" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 200**2+23.54**2-180**2 }{ 2 * 200 * 23.54 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 146° 15'4" - 30° = 3° 44'56" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1176.94 }{ 201.77 } = 5.83 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 200 }{ 2 * sin 146° 15'4" } = 180 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 180**2+2 * 23.54**2 - 200**2 } }{ 2 } = 80.48 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.54**2+2 * 200**2 - 180**2 } }{ 2 } = 110.35 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 180**2+2 * 200**2 - 23.54**2 } }{ 2 } = 189.899 ; ;
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