Triangle calculator SSA

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Triangle has two solutions with side c=26.9065779993 and with side c=9.70105178838

#1 Acute scalene triangle.

Sides: a = 45   b = 42   c = 26.9065779993

Area: T = 553.0422194681
Perimeter: p = 113.9065779993
Semiperimeter: s = 56.95328899965

Angle ∠ A = α = 78.18107714952° = 78°10'51″ = 1.36545118743 rad
Angle ∠ B = β = 66° = 1.15219173063 rad
Angle ∠ C = γ = 35.81992285048° = 35°49'9″ = 0.62551634729 rad

Height: ha = 24.5879653097
Height: hb = 26.33553426039
Height: hc = 41.11095455939

Median: ma = 27.16108265433
Median: mb = 30.5532585791
Median: mc = 41.39546826385

Inradius: r = 9.7110520304
Circumradius: R = 22.98773618486

Vertex coordinates: A[26.9065779993; 0] B[0; 0] C[18.30331489384; 41.11095455939]
Centroid: CG[15.07696429771; 13.70331818646]
Coordinates of the circumscribed circle: U[13.45328899965; 18.64397037396]
Coordinates of the inscribed circle: I[14.95328899965; 9.7110520304]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 101.8199228505° = 101°49'9″ = 1.36545118743 rad
∠ B' = β' = 114° = 1.15219173063 rad
∠ C' = γ' = 144.1810771495° = 144°10'51″ = 0.62551634729 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 = 45 ; ; b = 42 ; ; c = 26.91 ; ;

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

p = a+b+c = 45+42+26.91 = 113.91 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 113.91 }{ 2 } = 56.95 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 56.95 * (56.95-45)(56.95-42)(56.95-26.91) } ; ; T = sqrt{ 305855.67 } = 553.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 553.04 }{ 45 } = 24.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 553.04 }{ 42 } = 26.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 553.04 }{ 26.91 } = 41.11 ; ;

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{ 45**2-42**2-26.91**2 }{ 2 * 42 * 26.91 } ) = 78° 10'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-45**2-26.91**2 }{ 2 * 45 * 26.91 } ) = 66° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 26.91**2-45**2-42**2 }{ 2 * 42 * 45 } ) = 35° 49'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 553.04 }{ 56.95 } = 9.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 78° 10'51" } = 22.99 ; ;





#2 Obtuse scalene triangle.

Sides: a = 45   b = 42   c = 9.70105178838

Area: T = 199.3921941115
Perimeter: p = 96.70105178838
Semiperimeter: s = 48.35502589419

Angle ∠ A = α = 101.8199228505° = 101°49'9″ = 1.77770807792 rad
Angle ∠ B = β = 66° = 1.15219173063 rad
Angle ∠ C = γ = 12.18107714952° = 12°10'51″ = 0.2132594568 rad

Height: ha = 8.86218640495
Height: hb = 9.49548543388
Height: hc = 41.11095455939

Median: ma = 20.5622101634
Median: mb = 24.87106659261
Median: mc = 43.25547683868

Inradius: r = 4.12439063756
Circumradius: R = 22.98773618486

Vertex coordinates: A[9.70105178838; 0] B[0; 0] C[18.30331489384; 41.11095455939]
Centroid: CG[9.33545556074; 13.70331818646]
Coordinates of the circumscribed circle: U[4.85502589419; 22.47698418543]
Coordinates of the inscribed circle: I[6.35502589419; 4.12439063756]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 78.18107714952° = 78°10'51″ = 1.77770807792 rad
∠ B' = β' = 114° = 1.15219173063 rad
∠ C' = γ' = 167.8199228505° = 167°49'9″ = 0.2132594568 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 45 ; ; b = 42 ; ; beta = 66° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 42**2 = 45**2 + c**2 -2 * 42 * c * cos (66° ) ; ; ; ; c**2 -36.606c +261 =0 ; ; p=1; q=-36.6062978768; r=261 ; ; D = q**2 - 4pr = 36.606**2 - 4 * 1 * 261 = 296.021044247 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 36.61 ± sqrt{ 296.02 } }{ 2 } ; ; c_{1,2} = 18.3031489384 ± 8.60263105461 ; ; c_{1} = 26.905779993 ; ;
c_{2} = 9.70051788381 ; ; ; ; (c -26.905779993) (c -9.70051788381) = 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 = 45 ; ; b = 42 ; ; c = 9.7 ; ;

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

p = a+b+c = 45+42+9.7 = 96.7 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 96.7 }{ 2 } = 48.35 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 48.35 * (48.35-45)(48.35-42)(48.35-9.7) } ; ; T = sqrt{ 39757.15 } = 199.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 199.39 }{ 45 } = 8.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 199.39 }{ 42 } = 9.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 199.39 }{ 9.7 } = 41.11 ; ;

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{ 45**2-42**2-9.7**2 }{ 2 * 42 * 9.7 } ) = 101° 49'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-45**2-9.7**2 }{ 2 * 45 * 9.7 } ) = 66° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.7**2-45**2-42**2 }{ 2 * 42 * 45 } ) = 12° 10'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 199.39 }{ 48.35 } = 4.12 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 101° 49'9" } = 22.99 ; ;




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