Triangle calculator SSA

Please enter two sides and a non-included angle
°


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?

1. Use Law of Cosines

a = 45 ; ; b = 41 ; ; beta = 54° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 41**2 = 45**2 + c**2 -2 * 45 * c * cos (54° ) ; ; ; ; c**2 -52.901c +344 =0 ; ; p=1; q=-52.901; 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} = 26.45033635 ± 18.8578973694 ; ; c_{1} = 45.3082337194 ; ;
c_{2} = 7.59243898059 ; ; ; ; (c -45.3082337194) (c -7.59243898059) = 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 = 45.31 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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**2-45.31**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" ; ;

7. Inradius

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

8. 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 - 2ac cos beta ; ; 41**2 = 45**2 + c**2 -2 * 45 * c * cos (54° ) ; ; ; ; c**2 -52.901c +344 =0 ; ; p=1; q=-52.901; 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} = 26.45033635 ± 18.8578973694 ; ; c_{1} = 45.3082337194 ; ; : Nr. 1
c_{2} = 7.59243898059 ; ; ; ; (c -45.3082337194) (c -7.59243898059) = 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 = 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 ; ;




Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.