Triangle calculator SSA

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Triangle has two solutions with side c=98.50882623162 and with side c=66.87876389422

#1 Obtuse scalene triangle.

Sides: a = 88   b = 34   c = 98.50882623162

Area: T = 1482.443963983
Perimeter: p = 220.5088262316
Semiperimeter: s = 110.2544131158

Angle ∠ A = α = 62.28796667993° = 62°16'47″ = 1.08769852427 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 97.72203332007° = 97°43'13″ = 1.70655415605 rad

Height: ha = 33.69218099961
Height: hb = 87.20223317547
Height: hc = 30.09877726127

Median: ma = 59.11095497553
Median: mb = 91.84219232827
Median: mc = 44.98992271979

Inradius: r = 13.44656607137
Circumradius: R = 49.70546748028

Vertex coordinates: A[98.50882623162; 0] B[0; 0] C[82.69329506292; 30.09877726127]
Centroid: CG[60.44004043151; 10.03325908709]
Coordinates of the circumscribed circle: U[49.25441311581; -6.67772195645]
Coordinates of the inscribed circle: I[76.25441311581; 13.44656607137]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.7220333201° = 117°43'13″ = 1.08769852427 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 82.28796667993° = 82°16'47″ = 1.70655415605 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 88 ; ; b = 34 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34**2 = 88**2 + c**2 -2 * 88 * c * cos (20° ) ; ; ; ; c**2 -165.386c +6588 =0 ; ; p=1; q=-165.386; r=6588 ; ; D = q**2 - 4pr = 165.386**2 - 4 * 1 * 6588 = 1000.49633503 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 165.39 ± sqrt{ 1000.5 } }{ 2 } ; ; c_{1,2} = 82.69295063 ± 15.815311687 ; ; c_{1} = 98.508262317 ; ; c_{2} = 66.877638943 ; ; ; ; text{ Factored form: } ; ; (c -98.508262317) (c -66.877638943) = 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 = 88 ; ; b = 34 ; ; c = 98.51 ; ;

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

p = a+b+c = 88+34+98.51 = 220.51 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 220.51 }{ 2 } = 110.25 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.25 * (110.25-88)(110.25-34)(110.25-98.51) } ; ; T = sqrt{ 2197627.29 } = 1482.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1482.44 }{ 88 } = 33.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1482.44 }{ 34 } = 87.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1482.44 }{ 98.51 } = 30.1 ; ;

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{ 34**2+98.51**2-88**2 }{ 2 * 34 * 98.51 } ) = 62° 16'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 88**2+98.51**2-34**2 }{ 2 * 88 * 98.51 } ) = 20° ; ; gamma = 180° - alpha - beta = 180° - 62° 16'47" - 20° = 97° 43'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1482.44 }{ 110.25 } = 13.45 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 88 }{ 2 * sin 62° 16'47" } = 49.7 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 98.51**2 - 88**2 } }{ 2 } = 59.11 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 98.51**2+2 * 88**2 - 34**2 } }{ 2 } = 91.842 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 88**2 - 98.51**2 } }{ 2 } = 44.989 ; ;







#2 Obtuse scalene triangle.

Sides: a = 88   b = 34   c = 66.87876389422

Area: T = 1006.434398488
Perimeter: p = 188.8787638942
Semiperimeter: s = 94.43988194711

Angle ∠ A = α = 117.7220333201° = 117°43'13″ = 2.05546074109 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 42.28796667993° = 42°16'47″ = 0.73879193923 rad

Height: ha = 22.87334996563
Height: hb = 59.20219991104
Height: hc = 30.09877726127

Median: ma = 29.63662834249
Median: mb = 76.28443974561
Median: mc = 57.72221391875

Inradius: r = 10.65769945549
Circumradius: R = 49.70546748028

Vertex coordinates: A[66.87876389422; 0] B[0; 0] C[82.69329506292; 30.09877726127]
Centroid: CG[49.85768631904; 10.03325908709]
Coordinates of the circumscribed circle: U[33.43988194711; 36.77549921772]
Coordinates of the inscribed circle: I[60.43988194711; 10.65769945549]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 62.28796667993° = 62°16'47″ = 2.05546074109 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 137.7220333201° = 137°43'13″ = 0.73879193923 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 88 ; ; b = 34 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 34**2 = 88**2 + c**2 -2 * 88 * c * cos (20° ) ; ; ; ; c**2 -165.386c +6588 =0 ; ; p=1; q=-165.386; r=6588 ; ; D = q**2 - 4pr = 165.386**2 - 4 * 1 * 6588 = 1000.49633503 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 165.39 ± sqrt{ 1000.5 } }{ 2 } ; ; c_{1,2} = 82.69295063 ± 15.815311687 ; ; c_{1} = 98.508262317 ; ; c_{2} = 66.877638943 ; ; ; ; text{ Factored form: } ; ; (c -98.508262317) (c -66.877638943) = 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 = 88 ; ; b = 34 ; ; c = 66.88 ; ;

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

p = a+b+c = 88+34+66.88 = 188.88 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 188.88 }{ 2 } = 94.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 94.44 * (94.44-88)(94.44-34)(94.44-66.88) } ; ; T = sqrt{ 1012909.37 } = 1006.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1006.43 }{ 88 } = 22.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1006.43 }{ 34 } = 59.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1006.43 }{ 66.88 } = 30.1 ; ;

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{ 34**2+66.88**2-88**2 }{ 2 * 34 * 66.88 } ) = 117° 43'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 88**2+66.88**2-34**2 }{ 2 * 88 * 66.88 } ) = 20° ; ; gamma = 180° - alpha - beta = 180° - 117° 43'13" - 20° = 42° 16'47" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1006.43 }{ 94.44 } = 10.66 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 88 }{ 2 * sin 117° 43'13" } = 49.7 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 66.88**2 - 88**2 } }{ 2 } = 29.636 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 66.88**2+2 * 88**2 - 34**2 } }{ 2 } = 76.284 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 88**2 - 66.88**2 } }{ 2 } = 57.722 ; ;
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