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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 8.9**2 = 13.67**2 + c**2 -2 * 8.9 * c * cos (19° ) ; ; ; ; c**2 -25.85c +107.659 =0 ; ; p=1; q=-25.8504778569; r=107.6589 ; ; 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} = 12.9252389284 ± 7.7073277702 ; ;
c_{1} = 20.6325666986 ; ; c_{2} = 5.21791115824 ; ; ; ; (c -20.6325666986) (c -5.21791115824) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 13.67**2-8.9**2-5.22**2 }{ 2 * 8.9 * 5.22 } ) = 149° 59'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.9**2-13.67**2-5.22**2 }{ 2 * 13.67 * 5.22 } ) = 19° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.22**2-13.67**2-8.9**2 }{ 2 * 8.9 * 13.67 } ) = 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 ; ;




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