Triangle calculator SSA

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Triangle has two solutions with side c=25.31333134204 and with side c=5.37326668548

#1 Acute scalene triangle.

Sides: a = 35   b = 33   c = 25.31333134204

Area: T = 398.1550469401
Perimeter: p = 93.31333134204
Semiperimeter: s = 46.65766567102

Angle ∠ A = α = 72.41443585092° = 72°24'52″ = 1.26438689817 rad
Angle ∠ B = β = 64° = 1.11770107213 rad
Angle ∠ C = γ = 43.58656414908° = 43°35'8″ = 0.76107129506 rad

Height: ha = 22.75114553944
Height: hb = 24.13303314789
Height: hc = 31.45877916205

Median: ma = 23.63553954517
Median: mb = 25.70327609054
Median: mc = 31.57222827955

Inradius: r = 8.53436262277
Circumradius: R = 18.35879320178

Vertex coordinates: A[25.31333134204; 0] B[0; 0] C[15.34329901376; 31.45877916205]
Centroid: CG[13.5522101186; 10.48659305402]
Coordinates of the circumscribed circle: U[12.65766567102; 13.29774700185]
Coordinates of the inscribed circle: I[13.65766567102; 8.53436262277]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.5865641491° = 107°35'8″ = 1.26438689817 rad
∠ B' = β' = 116° = 1.11770107213 rad
∠ C' = γ' = 136.4144358509° = 136°24'52″ = 0.76107129506 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 33 ; ; beta = 64° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 35**2 + c**2 -2 * 35 * c * cos (64° ) ; ; ; ; c**2 -30.686c +136 =0 ; ; p=1; q=-30.686; r=136 ; ; D = q**2 - 4pr = 30.686**2 - 4 * 1 * 136 = 397.629385452 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.69 ± sqrt{ 397.63 } }{ 2 } ; ; c_{1,2} = 15.34299014 ± 9.97032328277 ; ; c_{1} = 25.3133134228 ; ;
c_{2} = 5.37266685723 ; ; ; ; text{ Factored form: } ; ; (c -25.3133134228) (c -5.37266685723) = 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 = 35 ; ; b = 33 ; ; c = 25.31 ; ;

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

p = a+b+c = 35+33+25.31 = 93.31 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 93.31 }{ 2 } = 46.66 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 46.66 * (46.66-35)(46.66-33)(46.66-25.31) } ; ; T = sqrt{ 158523.8 } = 398.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 398.15 }{ 35 } = 22.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 398.15 }{ 33 } = 24.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 398.15 }{ 25.31 } = 31.46 ; ;

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{ 33**2+25.31**2-35**2 }{ 2 * 33 * 25.31 } ) = 72° 24'52" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+25.31**2-33**2 }{ 2 * 35 * 25.31 } ) = 64° ; ; gamma = 180° - alpha - beta = 180° - 72° 24'52" - 64° = 43° 35'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 398.15 }{ 46.66 } = 8.53 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 72° 24'52" } = 18.36 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 25.31**2 - 35**2 } }{ 2 } = 23.635 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.31**2+2 * 35**2 - 33**2 } }{ 2 } = 25.703 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 35**2 - 25.31**2 } }{ 2 } = 31.572 ; ;





#2 Obtuse scalene triangle.

Sides: a = 35   b = 33   c = 5.37326668548

Area: T = 84.50661171829
Perimeter: p = 73.37326668548
Semiperimeter: s = 36.68663334274

Angle ∠ A = α = 107.5865641491° = 107°35'8″ = 1.87877236719 rad
Angle ∠ B = β = 64° = 1.11770107213 rad
Angle ∠ C = γ = 8.41443585092° = 8°24'52″ = 0.14768582604 rad

Height: ha = 4.82989209819
Height: hb = 5.12215828596
Height: hc = 31.45877916205

Median: ma = 15.89659986967
Median: mb = 18.8333023511
Median: mc = 33.90884593091

Inradius: r = 2.30334767797
Circumradius: R = 18.35879320178

Vertex coordinates: A[5.37326668548; 0] B[0; 0] C[15.34329901376; 31.45877916205]
Centroid: CG[6.90552189975; 10.48659305402]
Coordinates of the circumscribed circle: U[2.68663334274; 18.1660321602]
Coordinates of the inscribed circle: I[3.68663334274; 2.30334767797]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.41443585092° = 72°24'52″ = 1.87877236719 rad
∠ B' = β' = 116° = 1.11770107213 rad
∠ C' = γ' = 171.5865641491° = 171°35'8″ = 0.14768582604 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 33 ; ; beta = 64° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 35**2 + c**2 -2 * 35 * c * cos (64° ) ; ; ; ; c**2 -30.686c +136 =0 ; ; p=1; q=-30.686; r=136 ; ; D = q**2 - 4pr = 30.686**2 - 4 * 1 * 136 = 397.629385452 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 30.69 ± sqrt{ 397.63 } }{ 2 } ; ; c_{1,2} = 15.34299014 ± 9.97032328277 ; ; c_{1} = 25.3133134228 ; ; : Nr. 1
c_{2} = 5.37266685723 ; ; ; ; text{ Factored form: } ; ; (c -25.3133134228) (c -5.37266685723) = 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 = 35 ; ; b = 33 ; ; c = 5.37 ; ;

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

p = a+b+c = 35+33+5.37 = 73.37 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73.37 }{ 2 } = 36.69 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.69 * (36.69-35)(36.69-33)(36.69-5.37) } ; ; T = sqrt{ 7141.28 } = 84.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 84.51 }{ 35 } = 4.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 84.51 }{ 33 } = 5.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 84.51 }{ 5.37 } = 31.46 ; ;

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{ 33**2+5.37**2-35**2 }{ 2 * 33 * 5.37 } ) = 107° 35'8" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+5.37**2-33**2 }{ 2 * 35 * 5.37 } ) = 64° ; ; gamma = 180° - alpha - beta = 180° - 107° 35'8" - 64° = 8° 24'52" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 84.51 }{ 36.69 } = 2.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 107° 35'8" } = 18.36 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 5.37**2 - 35**2 } }{ 2 } = 15.896 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.37**2+2 * 35**2 - 33**2 } }{ 2 } = 18.833 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 35**2 - 5.37**2 } }{ 2 } = 33.908 ; ;
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