Triangle calculator SSA

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: h_a = 13.00113507617
Height: h_b = 17.3510893562
Height: h_c = 8.2498837187

Median: m_a = 24.843314153
Median: m_b = 27.86985213293
Median: m_c = 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?

1. Use Law of Cosines

$a = 22.02 ; ; b = 16.5 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.5**2 = 22.02**2 + c**2 -2 * 22.02 * c * cos (22° ) ; ; ; ; c**2 -40.833c +212.63 =0 ; ; p=1; q=-40.833; r=212.63 ; ; D = q**2 - 4pr = 40.833**2 - 4 * 1 * 212.63 = 816.826740248 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 40.83 ± sqrt{ 816.83 } }{ 2 } ; ; c_{1,2} = 20.41658848 ± 14.2900904498 ; ; c_{1} = 34.7066789298 ; ; c_{2} = 6.12649803025 ; ; ; ; text{ Factored form: } ; ; (c -34.7066789298) (c -6.12649803025) = 0 ; ; ; ; c>0 ; ;$

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.

$a = 22.02 ; ; b = 16.5 ; ; c = 34.71 ; ;$

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

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

3. Semiperimeter of the triangle

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

4. The triangle area using Heron's formula

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

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

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

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 16.5**2+34.71**2-22.02**2 }{ 2 * 16.5 * 34.71 } ) = 29° 59'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.02**2+34.71**2-16.5**2 }{ 2 * 22.02 * 34.71 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 29° 59'43" - 22° = 128° 17" ; ;$

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.5**2+2 * 34.71**2 - 22.02**2 } }{ 2 } = 24.843 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.71**2+2 * 22.02**2 - 16.5**2 } }{ 2 } = 27.869 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.5**2+2 * 22.02**2 - 34.71**2 } }{ 2 } = 8.799 ; ;$

#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: h_a = 2.29550265558
Height: h_b = 3.06328172581
Height: h_c = 8.2498837187

Median: m_a = 5.80327484042
Median: m_b = 13.89876504864
Median: m_c = 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

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How did we calculate this triangle?

1. Use Law of Cosines

$a = 22.02 ; ; b = 16.5 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.5**2 = 22.02**2 + c**2 -2 * 22.02 * c * cos (22° ) ; ; ; ; c**2 -40.833c +212.63 =0 ; ; p=1; q=-40.833; r=212.63 ; ; D = q**2 - 4pr = 40.833**2 - 4 * 1 * 212.63 = 816.826740248 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 40.83 ± sqrt{ 816.83 } }{ 2 } ; ; c_{1,2} = 20.41658848 ± 14.2900904498 ; ; c_{1} = 34.7066789298 ; ; c_{2} = 6.12649803025 ; ; ; ; text{ Factored form: } ; ; (c -34.7066789298) (c -6.12649803025) = 0 ; ; ; ; c>0 ; ; : Nr. 1$

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.

$a = 22.02 ; ; b = 16.5 ; ; c = 6.13 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 16.5**2+6.13**2-22.02**2 }{ 2 * 16.5 * 6.13 } ) = 150° 17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.02**2+6.13**2-16.5**2 }{ 2 * 22.02 * 6.13 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 150° 17" - 22° = 7° 59'43" ; ;$

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.5**2+2 * 6.13**2 - 22.02**2 } }{ 2 } = 5.803 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.13**2+2 * 22.02**2 - 16.5**2 } }{ 2 } = 13.898 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.5**2+2 * 22.02**2 - 6.13**2 } }{ 2 } = 19.214 ; ;$
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