Triangle calculator SSA

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Triangle has two solutions with side c=34.70766789273 and with side c=6.12664980278

#1 Obtuse scalene triangle.

Sides: a = 22.02   b = 16.5   c = 34.70766789273

Area: T = 143.1454871887
Perimeter: p = 73.22766789273
Semiperimeter: s = 36.61333394637

Angle ∠ A = α = 29.99553376194° = 29°59'43″ = 0.52435174017 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 128.0054662381° = 128°17″ = 2.23441028164 rad

Height: ha = 13.00113507617
Height: hb = 17.3510893562
Height: hc = 8.2498837187

Median: ma = 24.843314153
Median: mb = 27.86985213293
Median: mc = 8.79992505055

Inradius: r = 3.91096371427
Circumradius: R = 22.02331040911

Vertex coordinates: A[34.70766789273; 0] B[0; 0] C[20.41765884776; 8.2498837187]
Centroid: CG[18.37444224683; 2.75496123957]
Coordinates of the circumscribed circle: U[17.35333394637; -13.56601889096]
Coordinates of the inscribed circle: I[20.11333394637; 3.91096371427]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0054662381° = 150°17″ = 0.52435174017 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 51.99553376194° = 51°59'43″ = 2.23441028164 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 = 22.02 ; ; b = 16.5 ; ; c = 34.71 ; ;

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

p = a+b+c = 22.02+16.5+34.71 = 73.23 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73.23 }{ 2 } = 36.61 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.61 * (36.61-22.02)(36.61-16.5)(36.61-34.71) } ; ; T = sqrt{ 20490.45 } = 143.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 143.14 }{ 22.02 } = 13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 143.14 }{ 16.5 } = 17.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 143.14 }{ 34.71 } = 8.25 ; ;

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{ 22.02**2-16.5**2-34.71**2 }{ 2 * 16.5 * 34.71 } ) = 29° 59'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.5**2-22.02**2-34.71**2 }{ 2 * 22.02 * 34.71 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 34.71**2-22.02**2-16.5**2 }{ 2 * 16.5 * 22.02 } ) = 128° 17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 143.14 }{ 36.61 } = 3.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22.02 }{ 2 * sin 29° 59'43" } = 22.02 ; ;





#2 Obtuse scalene triangle.

Sides: a = 22.02   b = 16.5   c = 6.12664980278

Area: T = 25.2688242379
Perimeter: p = 44.64664980278
Semiperimeter: s = 22.32332490139

Angle ∠ A = α = 150.0054662381° = 150°17″ = 2.61880752519 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 7.99553376194° = 7°59'43″ = 0.14395449663 rad

Height: ha = 2.29550265558
Height: hb = 3.06328172581
Height: hc = 8.2498837187

Median: ma = 5.80327484042
Median: mb = 13.89876504864
Median: mc = 19.21441017349

Inradius: r = 1.13219249435
Circumradius: R = 22.02331040911

Vertex coordinates: A[6.12664980278; 0] B[0; 0] C[20.41765884776; 8.2498837187]
Centroid: CG[8.84876955018; 2.75496123957]
Coordinates of the circumscribed circle: U[3.06332490139; 21.80990260967]
Coordinates of the inscribed circle: I[5.82332490139; 1.13219249435]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.99553376194° = 29°59'43″ = 2.61880752519 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 172.0054662381° = 172°17″ = 0.14395449663 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 22.02 ; ; b = 16.5 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 16.5**2 = 22.02**2 + c**2 -2 * 16.5 * c * cos (22° ) ; ; ; ; c**2 -40.833c +212.63 =0 ; ; p=1; q=-40.8331769551; r=212.6304 ; ; D = q**2 - 4pr = 40.833**2 - 4 * 1 * 212.63 = 816.826740248 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 40.83 ± sqrt{ 816.83 } }{ 2 } ; ; c_{1,2} = 20.4165884776 ± 14.2900904498 ; ;
c_{1} = 34.7066789273 ; ; c_{2} = 6.12649802781 ; ; ; ; (c -34.7066789273) (c -6.12649802781) = 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 = 22.02 ; ; b = 16.5 ; ; c = 6.13 ; ;

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

p = a+b+c = 22.02+16.5+6.13 = 44.65 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44.65 }{ 2 } = 22.32 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.32 * (22.32-22.02)(22.32-16.5)(22.32-6.13) } ; ; T = sqrt{ 638.48 } = 25.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.27 }{ 22.02 } = 2.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.27 }{ 16.5 } = 3.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.27 }{ 6.13 } = 8.25 ; ;

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{ 22.02**2-16.5**2-6.13**2 }{ 2 * 16.5 * 6.13 } ) = 150° 17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.5**2-22.02**2-6.13**2 }{ 2 * 22.02 * 6.13 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.13**2-22.02**2-16.5**2 }{ 2 * 16.5 * 22.02 } ) = 7° 59'43" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.27 }{ 22.32 } = 1.13 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22.02 }{ 2 * sin 150° 17" } = 22.02 ; ;




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