Triangle calculator SSA

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Triangle has two solutions with side c=45.30882337226 and with side c=7.59224389837

#1 Acute scalene triangle.

Sides: a = 45   b = 41   c = 45.30882337226

Area: T = 824.7440449
Perimeter: p = 131.3088233723
Semiperimeter: s = 65.65441168613

Angle ∠ A = α = 62.61662016892° = 62°36'58″ = 1.09328588846 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 63.38437983108° = 63°23'2″ = 1.10662559729 rad

Height: ha = 36.65551310667
Height: hb = 40.23112414146
Height: hc = 36.40657647469

Median: ma = 36.88772338558
Median: mb = 40.23326735568
Median: mc = 36.60331554546

Inradius: r = 12.56218999756
Circumradius: R = 25.33993935387

Vertex coordinates: A[45.30882337226; 0] B[0; 0] C[26.45503363532; 36.40657647469]
Centroid: CG[23.92195233586; 12.13552549156]
Coordinates of the circumscribed circle: U[22.65441168613; 11.35223501596]
Coordinates of the inscribed circle: I[24.65441168613; 12.56218999756]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.3843798311° = 117°23'2″ = 1.09328588846 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 116.6166201689° = 116°36'58″ = 1.10662559729 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 = 45 ; ; b = 41 ; ; c = 45.31 ; ;

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

p = a+b+c = 45+41+45.31 = 131.31 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 131.31 }{ 2 } = 65.65 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.65 * (65.65-45)(65.65-41)(65.65-45.31) } ; ; T = sqrt{ 680196.81 } = 824.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 824.74 }{ 45 } = 36.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 824.74 }{ 41 } = 40.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 824.74 }{ 45.31 } = 36.41 ; ;

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{ 45**2-41**2-45.31**2 }{ 2 * 41 * 45.31 } ) = 62° 36'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 41**2-45**2-45.31**2 }{ 2 * 45 * 45.31 } ) = 54° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 45.31**2-45**2-41**2 }{ 2 * 41 * 45 } ) = 63° 23'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 824.74 }{ 65.65 } = 12.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 62° 36'58" } = 25.34 ; ;





#2 Obtuse scalene triangle.

Sides: a = 45   b = 41   c = 7.59224389837

Area: T = 138.2044273749
Perimeter: p = 93.59224389837
Semiperimeter: s = 46.79662194919

Angle ∠ A = α = 117.3843798311° = 117°23'2″ = 2.0498733769 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 8.61662016892° = 8°36'58″ = 0.15503810885 rad

Height: ha = 6.14224121666
Height: hb = 6.74216718902
Height: hc = 36.40657647469

Median: ma = 19.05444631218
Median: mb = 24.921132751
Median: mc = 42.87987676778

Inradius: r = 2.9533321342
Circumradius: R = 25.33993935387

Vertex coordinates: A[7.59224389837; 0] B[0; 0] C[26.45503363532; 36.40657647469]
Centroid: CG[11.3487591779; 12.13552549156]
Coordinates of the circumscribed circle: U[3.79662194919; 25.05334145873]
Coordinates of the inscribed circle: I[5.79662194919; 2.9533321342]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 62.61662016892° = 62°36'58″ = 2.0498733769 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 171.3843798311° = 171°23'2″ = 0.15503810885 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 45 ; ; b = 41 ; ; beta = 54° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 41**2 = 45**2 + c**2 -2 * 41 * c * cos (54° ) ; ; ; ; c**2 -52.901c +344 =0 ; ; p=1; q=-52.9006727063; r=344 ; ; D = q**2 - 4pr = 52.901**2 - 4 * 1 * 344 = 1422.48117278 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 52.9 ± sqrt{ 1422.48 } }{ 2 } ; ; c_{1,2} = 26.4503363532 ± 18.8578973694 ; ; c_{1} = 45.3082337226 ; ;
c_{2} = 7.59243898375 ; ; ; ; (c -45.3082337226) (c -7.59243898375) = 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 = 45 ; ; b = 41 ; ; c = 7.59 ; ;

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

p = a+b+c = 45+41+7.59 = 93.59 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 93.59 }{ 2 } = 46.8 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 46.8 * (46.8-45)(46.8-41)(46.8-7.59) } ; ; T = sqrt{ 19100.42 } = 138.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 138.2 }{ 45 } = 6.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 138.2 }{ 41 } = 6.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 138.2 }{ 7.59 } = 36.41 ; ;

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{ 45**2-41**2-7.59**2 }{ 2 * 41 * 7.59 } ) = 117° 23'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 41**2-45**2-7.59**2 }{ 2 * 45 * 7.59 } ) = 54° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.59**2-45**2-41**2 }{ 2 * 41 * 45 } ) = 8° 36'58" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 138.2 }{ 46.8 } = 2.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45 }{ 2 * sin 117° 23'2" } = 25.34 ; ;




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