Triangle calculator SSA

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Triangle has two solutions with side c=74.98797353557 and with side c=4.69546017925

#1 Obtuse scalene triangle.

Sides: a = 46   b = 42   c = 74.98797353557

Area: T = 862.267695659
Perimeter: p = 162.9879735356
Semiperimeter: s = 81.49898676778

Angle ∠ A = α = 33.204382253° = 33°12'14″ = 0.58795160274 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 116.796617747° = 116°47'46″ = 2.03884778506 rad

Height: ha = 37.49898676778
Height: hb = 41.06603312662
Height: hc = 23

Median: ma = 56.24992698353
Median: mb = 58.5498956925
Median: mc = 23.11994684519

Inradius: r = 10.58112781535
Circumradius: R = 42

Vertex coordinates: A[74.98797353557; 0] B[0; 0] C[39.83771685741; 23]
Centroid: CG[38.27223013099; 7.66766666667]
Coordinates of the circumscribed circle: U[37.49898676778; -18.93443555871]
Coordinates of the inscribed circle: I[39.49898676778; 10.58112781535]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.796617747° = 146°47'46″ = 0.58795160274 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 63.204382253° = 63°12'14″ = 2.03884778506 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 46 ; ; b = 42 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 46**2 + c**2 -2 * 46 * c * cos (30° ) ; ; ; ; c**2 -79.674c +352 =0 ; ; p=1; q=-79.674; r=352 ; ; D = q**2 - 4pr = 79.674**2 - 4 * 1 * 352 = 4940 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 79.67 ± sqrt{ 4940 } }{ 2 } ; ; c_{1,2} = 39.83716857 ± 35.1425667816 ; ; c_{1} = 74.9797353516 ; ; c_{2} = 4.69460178839 ; ; ; ; text{ Factored form: } ; ; (c -74.9797353516) (c -4.69460178839) = 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 = 46 ; ; b = 42 ; ; c = 74.98 ; ;

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

p = a+b+c = 46+42+74.98 = 162.98 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 162.98 }{ 2 } = 81.49 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.49 * (81.49-46)(81.49-42)(81.49-74.98) } ; ; T = sqrt{ 743504.3 } = 862.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 862.27 }{ 46 } = 37.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 862.27 }{ 42 } = 41.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 862.27 }{ 74.98 } = 23 ; ;

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{ 42**2+74.98**2-46**2 }{ 2 * 42 * 74.98 } ) = 33° 12'14" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+74.98**2-42**2 }{ 2 * 46 * 74.98 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 33° 12'14" - 30° = 116° 47'46" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 862.27 }{ 81.49 } = 10.58 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46 }{ 2 * sin 33° 12'14" } = 42 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 74.98**2 - 46**2 } }{ 2 } = 56.249 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 74.98**2+2 * 46**2 - 42**2 } }{ 2 } = 58.549 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 46**2 - 74.98**2 } }{ 2 } = 23.119 ; ;







#2 Obtuse scalene triangle.

Sides: a = 46   b = 42   c = 4.69546017925

Area: T = 53.98879206134
Perimeter: p = 92.69546017925
Semiperimeter: s = 46.34773008962

Angle ∠ A = α = 146.796617747° = 146°47'46″ = 2.56220766262 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 3.204382253° = 3°12'14″ = 0.05659172518 rad

Height: ha = 2.34773008962
Height: hb = 2.57108533625
Height: hc = 23

Median: ma = 19.07992988077
Median: mb = 25.06603200896
Median: mc = 43.98328395912

Inradius: r = 1.16548557644
Circumradius: R = 42

Vertex coordinates: A[4.69546017925; 0] B[0; 0] C[39.83771685741; 23]
Centroid: CG[14.84439234555; 7.66766666667]
Coordinates of the circumscribed circle: U[2.34773008962; 41.93443555871]
Coordinates of the inscribed circle: I[4.34773008962; 1.16548557644]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 33.204382253° = 33°12'14″ = 2.56220766262 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 176.796617747° = 176°47'46″ = 0.05659172518 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 46 ; ; b = 42 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 46**2 + c**2 -2 * 46 * c * cos (30° ) ; ; ; ; c**2 -79.674c +352 =0 ; ; p=1; q=-79.674; r=352 ; ; D = q**2 - 4pr = 79.674**2 - 4 * 1 * 352 = 4940 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 79.67 ± sqrt{ 4940 } }{ 2 } ; ; c_{1,2} = 39.83716857 ± 35.1425667816 ; ; c_{1} = 74.9797353516 ; ; c_{2} = 4.69460178839 ; ; ; ; text{ Factored form: } ; ; (c -74.9797353516) (c -4.69460178839) = 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 = 46 ; ; b = 42 ; ; c = 4.69 ; ;

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

p = a+b+c = 46+42+4.69 = 92.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 92.69 }{ 2 } = 46.35 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 46.35 * (46.35-46)(46.35-42)(46.35-4.69) } ; ; T = sqrt{ 2914.7 } = 53.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.99 }{ 46 } = 2.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.99 }{ 42 } = 2.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.99 }{ 4.69 } = 23 ; ;

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{ 42**2+4.69**2-46**2 }{ 2 * 42 * 4.69 } ) = 146° 47'46" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+4.69**2-42**2 }{ 2 * 46 * 4.69 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 146° 47'46" - 30° = 3° 12'14" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.99 }{ 46.35 } = 1.16 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46 }{ 2 * sin 146° 47'46" } = 42 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 4.69**2 - 46**2 } }{ 2 } = 19.079 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.69**2+2 * 46**2 - 42**2 } }{ 2 } = 25.06 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 46**2 - 4.69**2 } }{ 2 } = 43.983 ; ;
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