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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 16.15**2 = 18.76**2 + c**2 -2 * 16.15 * c * cos (35° 30') ; ; ; ; c**2 -30.546c +91.115 =0 ; ; p=1; q=-30.5456142487; r=91.1151 ; ; 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} = 15.2728071244 ± 11.9223964646 ; ;
c_{1} = 27.1952035889 ; ; c_{2} = 3.35041065981 ; ; ; ; (c -27.1952035889) (c -3.35041065981) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 18.76**2-16.15**2-3.35**2 }{ 2 * 16.15 * 3.35 } ) = 137° 34'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.15**2-18.76**2-3.35**2 }{ 2 * 18.76 * 3.35 } ) = 35° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.35**2-18.76**2-16.15**2 }{ 2 * 16.15 * 18.76 } ) = 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 ; ;




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