Triangle calculator SSA

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Triangle has two solutions with side c=46.15224495884 and with side c=8.84402674969

#1 Acute scalene triangle.

Sides: a = 37   b = 31   c = 46.15224495884

Area: T = 571.3177306694
Perimeter: p = 114.1522449588
Semiperimeter: s = 57.07662247942

Angle ∠ A = α = 533.0004042355° = 53°1″ = 0.92550315588 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 854.9995957645° = 84°59'59″ = 1.4843522809 rad

Height: ha = 30.8822016578
Height: hb = 36.8599181077
Height: hc = 24.75878324353

Median: ma = 34.68882444281
Median: mb = 38.84993796798
Median: mc = 25.14993111088

Inradius: r = 10.0109724868
Circumradius: R = 23.16443865229

Vertex coordinates: A[46.15224495884; 0] B[0; 0] C[27.49663585427; 24.75878324353]
Centroid: CG[24.55496027104; 8.25326108118]
Coordinates of the circumscribed circle: U[23.07662247942; 2.0199072121]
Coordinates of the inscribed circle: I[26.07662247942; 10.0109724868]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 1276.999595765° = 126°59'59″ = 0.92550315588 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 955.0004042355° = 95°1″ = 1.4843522809 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 = 37 ; ; b = 31 ; ; c = 46.15 ; ;

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

p = a+b+c = 37+31+46.15 = 114.15 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 114.15 }{ 2 } = 57.08 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.08 * (57.08-37)(57.08-31)(57.08-46.15) } ; ; T = sqrt{ 326403.46 } = 571.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 571.32 }{ 37 } = 30.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 571.32 }{ 31 } = 36.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 571.32 }{ 46.15 } = 24.76 ; ;

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{ 37**2-31**2-46.15**2 }{ 2 * 31 * 46.15 } ) = 53° 1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-37**2-46.15**2 }{ 2 * 37 * 46.15 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 46.15**2-37**2-31**2 }{ 2 * 31 * 37 } ) = 84° 59'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 571.32 }{ 57.08 } = 10.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37 }{ 2 * sin 53° 1" } = 23.16 ; ;





#2 Obtuse scalene triangle.

Sides: a = 37   b = 31   c = 8.84402674969

Area: T = 109.4332930686
Perimeter: p = 76.84402674969
Semiperimeter: s = 38.42201337485

Angle ∠ A = α = 1276.999595765° = 126°59'59″ = 2.21765610948 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 111.0004042355° = 11°1″ = 0.1921993273 rad

Height: ha = 5.91552935506
Height: hb = 7.06601890765
Height: hc = 24.75878324353

Median: ma = 13.31663495264
Median: mb = 21.98546574845
Median: mc = 33.84546807881

Inradius: r = 2.84883224812
Circumradius: R = 23.16443865229

Vertex coordinates: A[8.84402674969; 0] B[0; 0] C[27.49663585427; 24.75878324353]
Centroid: CG[12.11222086799; 8.25326108118]
Coordinates of the circumscribed circle: U[4.42201337485; 22.73987603142]
Coordinates of the inscribed circle: I[7.42201337485; 2.84883224812]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 533.0004042355° = 53°1″ = 2.21765610948 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 1698.999595765° = 168°59'59″ = 0.1921993273 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 37 ; ; b = 31 ; ; beta = 42° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 31**2 = 37**2 + c**2 -2 * 31 * c * cos (42° ) ; ; ; ; c**2 -54.993c +408 =0 ; ; p=1; q=-54.9927170853; r=408 ; ; D = q**2 - 4pr = 54.993**2 - 4 * 1 * 408 = 1392.19893243 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 54.99 ± sqrt{ 1392.2 } }{ 2 } ; ; c_{1,2} = 27.4963585427 ± 18.6560910457 ; ; c_{1} = 46.1524495884 ; ;
c_{2} = 8.84026749693 ; ; ; ; (c -46.1524495884) (c -8.84026749693) = 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 = 37 ; ; b = 31 ; ; c = 8.84 ; ;

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

p = a+b+c = 37+31+8.84 = 76.84 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76.84 }{ 2 } = 38.42 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.42 * (38.42-37)(38.42-31)(38.42-8.84) } ; ; T = sqrt{ 11975.57 } = 109.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.43 }{ 37 } = 5.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.43 }{ 31 } = 7.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.43 }{ 8.84 } = 24.76 ; ;

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{ 37**2-31**2-8.84**2 }{ 2 * 31 * 8.84 } ) = 126° 59'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-37**2-8.84**2 }{ 2 * 37 * 8.84 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.84**2-37**2-31**2 }{ 2 * 31 * 37 } ) = 11° 1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.43 }{ 38.42 } = 2.85 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37 }{ 2 * sin 126° 59'59" } = 23.16 ; ;




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