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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 88 ; ; b = 34 ; ; beta = 20° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 34**2 = 88**2 + c**2 -2 * 34 * c * cos (20° ) ; ; ; ; c**2 -165.386c +6588 =0 ; ; p=1; q=-165.385901258; 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} = 82.6929506292 ± 15.815311687 ; ; c_{1} = 98.5082623162 ; ;
c_{2} = 66.8776389422 ; ; ; ; (c -98.5082623162) (c -66.8776389422) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 88**2-34**2-66.88**2 }{ 2 * 34 * 66.88 } ) = 117° 43'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 34**2-88**2-66.88**2 }{ 2 * 88 * 66.88 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 66.88**2-88**2-34**2 }{ 2 * 34 * 88 } ) = 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 ; ;




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