Triangle calculator SSA

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Triangle has two solutions with side c=20.63325666986 and with side c=5.21879111582

#1 Obtuse scalene triangle.

Sides: a = 13.67   b = 8.9   c = 20.63325666986

Area: T = 45.91327910333
Perimeter: p = 43.20325666986
Semiperimeter: s = 21.60112833493

Angle ∠ A = α = 30.00438408279° = 30°14″ = 0.52436658107 rad
Angle ∠ B = β = 19° = 0.33216125579 rad
Angle ∠ C = γ = 130.9966159172° = 130°59'46″ = 2.2866314285 rad

Height: ha = 6.71773066618
Height: hb = 10.31774811311
Height: hc = 4.45105166714

Median: ma = 14.34436110965
Median: mb = 16.92658191615
Median: mc = 5.15988514086

Inradius: r = 2.12554658944
Circumradius: R = 13.66884130161

Vertex coordinates: A[20.63325666986; 0] B[0; 0] C[12.92552389284; 4.45105166714]
Centroid: CG[11.1865935209; 1.48435055571]
Coordinates of the circumscribed circle: U[10.31662833493; -8.96765942383]
Coordinates of the inscribed circle: I[12.70112833493; 2.12554658944]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.9966159172° = 149°59'46″ = 0.52436658107 rad
∠ B' = β' = 161° = 0.33216125579 rad
∠ C' = γ' = 49.00438408279° = 49°14″ = 2.2866314285 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 13.67 ; ; b = 8.9 ; ; beta = 19° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.9**2 = 13.67**2 + c**2 -2 * 13.67 * c * cos (19° ) ; ; ; ; c**2 -25.85c +107.659 =0 ; ; p=1; q=-25.85; r=107.659 ; ; D = q**2 - 4pr = 25.85**2 - 4 * 1 * 107.659 = 237.611605429 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 25.85 ± sqrt{ 237.61 } }{ 2 } ; ; c_{1,2} = 12.92523893 ± 7.7073277702 ; ; c_{1} = 20.6325667002 ; ; c_{2} = 5.2179111598 ; ; ; ; text{ Factored form: } ; ; (c -20.6325667002) (c -5.2179111598) = 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 = 13.67 ; ; b = 8.9 ; ; c = 20.63 ; ;

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

p = a+b+c = 13.67+8.9+20.63 = 43.2 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43.2 }{ 2 } = 21.6 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.6 * (21.6-13.67)(21.6-8.9)(21.6-20.63) } ; ; T = sqrt{ 2107.98 } = 45.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45.91 }{ 13.67 } = 6.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45.91 }{ 8.9 } = 10.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45.91 }{ 20.63 } = 4.45 ; ;

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{ 8.9**2+20.63**2-13.67**2 }{ 2 * 8.9 * 20.63 } ) = 30° 14" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.67**2+20.63**2-8.9**2 }{ 2 * 13.67 * 20.63 } ) = 19° ; ; gamma = 180° - alpha - beta = 180° - 30° 14" - 19° = 130° 59'46" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45.91 }{ 21.6 } = 2.13 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.67 }{ 2 * sin 30° 14" } = 13.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.9**2+2 * 20.63**2 - 13.67**2 } }{ 2 } = 14.344 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.63**2+2 * 13.67**2 - 8.9**2 } }{ 2 } = 16.926 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.9**2+2 * 13.67**2 - 20.63**2 } }{ 2 } = 5.159 ; ;







#2 Obtuse scalene triangle.

Sides: a = 13.67   b = 8.9   c = 5.21879111582

Area: T = 11.61112002999
Perimeter: p = 27.78879111582
Semiperimeter: s = 13.89439555791

Angle ∠ A = α = 149.9966159172° = 149°59'46″ = 2.61879268429 rad
Angle ∠ B = β = 19° = 0.33216125579 rad
Angle ∠ C = γ = 11.00438408279° = 11°14″ = 0.19220532528 rad

Height: ha = 1.69987857059
Height: hb = 2.60992584944
Height: hc = 4.45105166714

Median: ma = 2.55497202646
Median: mb = 9.34105164968
Median: mc = 11.23553371461

Inradius: r = 0.83657015562
Circumradius: R = 13.66884130161

Vertex coordinates: A[5.21879111582; 0] B[0; 0] C[12.92552389284; 4.45105166714]
Centroid: CG[6.04877166956; 1.48435055571]
Coordinates of the circumscribed circle: U[2.60989555791; 13.41771109097]
Coordinates of the inscribed circle: I[4.99439555791; 0.83657015562]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.00438408279° = 30°14″ = 2.61879268429 rad
∠ B' = β' = 161° = 0.33216125579 rad
∠ C' = γ' = 168.9966159172° = 168°59'46″ = 0.19220532528 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 13.67 ; ; b = 8.9 ; ; beta = 19° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.9**2 = 13.67**2 + c**2 -2 * 13.67 * c * cos (19° ) ; ; ; ; c**2 -25.85c +107.659 =0 ; ; p=1; q=-25.85; r=107.659 ; ; D = q**2 - 4pr = 25.85**2 - 4 * 1 * 107.659 = 237.611605429 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 25.85 ± sqrt{ 237.61 } }{ 2 } ; ; c_{1,2} = 12.92523893 ± 7.7073277702 ; ; c_{1} = 20.6325667002 ; ; c_{2} = 5.2179111598 ; ; ; ; text{ Factored form: } ; ; (c -20.6325667002) (c -5.2179111598) = 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 = 13.67 ; ; b = 8.9 ; ; c = 5.22 ; ;

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

p = a+b+c = 13.67+8.9+5.22 = 27.79 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.79 }{ 2 } = 13.89 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.89 * (13.89-13.67)(13.89-8.9)(13.89-5.22) } ; ; T = sqrt{ 134.82 } = 11.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.61 }{ 13.67 } = 1.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.61 }{ 8.9 } = 2.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.61 }{ 5.22 } = 4.45 ; ;

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{ 8.9**2+5.22**2-13.67**2 }{ 2 * 8.9 * 5.22 } ) = 149° 59'46" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.67**2+5.22**2-8.9**2 }{ 2 * 13.67 * 5.22 } ) = 19° ; ; gamma = 180° - alpha - beta = 180° - 149° 59'46" - 19° = 11° 14" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.61 }{ 13.89 } = 0.84 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.67 }{ 2 * sin 149° 59'46" } = 13.67 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.9**2+2 * 5.22**2 - 13.67**2 } }{ 2 } = 2.55 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.22**2+2 * 13.67**2 - 8.9**2 } }{ 2 } = 9.341 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.9**2+2 * 13.67**2 - 5.22**2 } }{ 2 } = 11.235 ; ;
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