Triangle calculator SSA

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Triangle has two solutions with side c=27.19552035889 and with side c=3.35504106598

#1 Obtuse scalene triangle.

Sides: a = 18.76   b = 16.15   c = 27.19552035889

Area: T = 148.1322103287
Perimeter: p = 62.10552035889
Semiperimeter: s = 31.05326017945

Angle ∠ A = α = 42.41992386486° = 42°25'9″ = 0.74403553806 rad
Angle ∠ B = β = 35.5° = 35°30' = 0.62195918845 rad
Angle ∠ C = γ = 102.0810761351° = 102°4'51″ = 1.78216453885 rad

Height: ha = 15.79223351052
Height: hb = 18.34545329148
Height: hc = 10.89439874491

Median: ma = 20.30331130402
Median: mb = 21.92215128155
Median: mc = 11.02220359026

Inradius: r = 4.77703604441
Circumradius: R = 13.90655603568

Vertex coordinates: A[27.19552035889; 0] B[0; 0] C[15.27328071244; 10.89439874491]
Centroid: CG[14.15660035711; 3.63113291497]
Coordinates of the circumscribed circle: U[13.59876017945; -2.91102979702]
Coordinates of the inscribed circle: I[14.90326017945; 4.77703604441]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.5810761351° = 137°34'51″ = 0.74403553806 rad
∠ B' = β' = 144.5° = 144°30' = 0.62195918845 rad
∠ C' = γ' = 77.91992386486° = 77°55'9″ = 1.78216453885 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 18.76 ; ; b = 16.15 ; ; beta = 35° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.15**2 = 18.76**2 + c**2 -2 * 18.76 * c * cos (35° 30') ; ; ; ; c**2 -30.546c +91.115 =0 ; ; p=1; q=-30.546; r=91.115 ; ; D = q**2 - 4pr = 30.546**2 - 4 * 1 * 91.115 = 568.574149832 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.55 ± sqrt{ 568.57 } }{ 2 } ; ; c_{1,2} = 15.27280712 ± 11.9223964646 ; ;
c_{1} = 27.1952035846 ; ; c_{2} = 3.35041065544 ; ; ; ; text{ Factored form: } ; ; (c -27.1952035846) (c -3.35041065544) = 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 = 18.76 ; ; b = 16.15 ; ; c = 27.2 ; ;

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

p = a+b+c = 18.76+16.15+27.2 = 62.11 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62.11 }{ 2 } = 31.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.05 * (31.05-18.76)(31.05-16.15)(31.05-27.2) } ; ; T = sqrt{ 21943.12 } = 148.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 148.13 }{ 18.76 } = 15.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 148.13 }{ 16.15 } = 18.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 148.13 }{ 27.2 } = 10.89 ; ;

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{ 16.15**2+27.2**2-18.76**2 }{ 2 * 16.15 * 27.2 } ) = 42° 25'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.76**2+27.2**2-16.15**2 }{ 2 * 18.76 * 27.2 } ) = 35° 30' ; ; gamma = 180° - alpha - beta = 180° - 42° 25'9" - 35° 30' = 102° 4'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 148.13 }{ 31.05 } = 4.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.76 }{ 2 * sin 42° 25'9" } = 13.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.15**2+2 * 27.2**2 - 18.76**2 } }{ 2 } = 20.303 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.2**2+2 * 18.76**2 - 16.15**2 } }{ 2 } = 21.922 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.15**2+2 * 18.76**2 - 27.2**2 } }{ 2 } = 11.022 ; ;







#2 Obtuse scalene triangle.

Sides: a = 18.76   b = 16.15   c = 3.35504106598

Area: T = 18.25496658387
Perimeter: p = 38.26604106598
Semiperimeter: s = 19.13302053299

Angle ∠ A = α = 137.5810761351° = 137°34'51″ = 2.4011237273 rad
Angle ∠ B = β = 35.5° = 35°30' = 0.62195918845 rad
Angle ∠ C = γ = 6.91992386486° = 6°55'9″ = 0.12107634961 rad

Height: ha = 1.9465593373
Height: hb = 2.26600205373
Height: hc = 10.89439874491

Median: ma = 6.93110515649
Median: mb = 10.78877616211
Median: mc = 17.42333675592

Inradius: r = 0.95439712472
Circumradius: R = 13.90655603568

Vertex coordinates: A[3.35504106598; 0] B[0; 0] C[15.27328071244; 10.89439874491]
Centroid: CG[6.20877392614; 3.63113291497]
Coordinates of the circumscribed circle: U[1.67552053299; 13.80442854193]
Coordinates of the inscribed circle: I[2.98802053299; 0.95439712472]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.41992386486° = 42°25'9″ = 2.4011237273 rad
∠ B' = β' = 144.5° = 144°30' = 0.62195918845 rad
∠ C' = γ' = 173.0810761351° = 173°4'51″ = 0.12107634961 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 18.76 ; ; b = 16.15 ; ; beta = 35° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.15**2 = 18.76**2 + c**2 -2 * 18.76 * c * cos (35° 30') ; ; ; ; c**2 -30.546c +91.115 =0 ; ; p=1; q=-30.546; r=91.115 ; ; D = q**2 - 4pr = 30.546**2 - 4 * 1 * 91.115 = 568.574149832 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.55 ± sqrt{ 568.57 } }{ 2 } ; ; c_{1,2} = 15.27280712 ± 11.9223964646 ; ; : Nr. 1
c_{1} = 27.1952035846 ; ; c_{2} = 3.35041065544 ; ; ; ; text{ Factored form: } ; ; (c -27.1952035846) (c -3.35041065544) = 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 = 18.76 ; ; b = 16.15 ; ; c = 3.35 ; ;

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

p = a+b+c = 18.76+16.15+3.35 = 38.26 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.26 }{ 2 } = 19.13 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.13 * (19.13-18.76)(19.13-16.15)(19.13-3.35) } ; ; T = sqrt{ 333.05 } = 18.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.25 }{ 18.76 } = 1.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.25 }{ 16.15 } = 2.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.25 }{ 3.35 } = 10.89 ; ;

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{ 16.15**2+3.35**2-18.76**2 }{ 2 * 16.15 * 3.35 } ) = 137° 34'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.76**2+3.35**2-16.15**2 }{ 2 * 18.76 * 3.35 } ) = 35° 30' ; ; gamma = 180° - alpha - beta = 180° - 137° 34'51" - 35° 30' = 6° 55'9" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.25 }{ 19.13 } = 0.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.76 }{ 2 * sin 137° 34'51" } = 13.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.15**2+2 * 3.35**2 - 18.76**2 } }{ 2 } = 6.931 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.35**2+2 * 18.76**2 - 16.15**2 } }{ 2 } = 10.788 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.15**2+2 * 18.76**2 - 3.35**2 } }{ 2 } = 17.423 ; ;
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