Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=19.42223257971 and with side c=16.70765470629

#1 Acute scalene triangle.

Sides: a = 24.7   b = 16.9   c = 19.42223257971

Area: T = 163.5888030126
Perimeter: p = 61.02223257971
Semiperimeter: s = 30.51111628985

Angle ∠ A = α = 85.39114031612° = 85°23'29″ = 1.49903611381 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 51.60985968388° = 51°36'31″ = 0.90107399372 rad

Height: ha = 13.24659943422
Height: hb = 19.36595301925
Height: hc = 16.84553594935

Median: ma = 13.37551960615
Median: mb = 20.54988654111
Median: mc = 18.80327475428

Inradius: r = 5.3621579651
Circumradius: R = 12.39900591187

Vertex coordinates: A[19.42223257971; 0] B[0; 0] C[18.064443643; 16.84553594935]
Centroid: CG[12.4965587409; 5.61551198312]
Coordinates of the circumscribed circle: U[9.71111628985; 7.69546007123]
Coordinates of the inscribed circle: I[13.61111628985; 5.3621579651]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 94.60985968388° = 94°36'31″ = 1.49903611381 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 128.3911403161° = 128°23'29″ = 0.90107399372 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 24.7 ; ; b = 16.9 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.9**2 = 24.7**2 + c**2 -2 * 24.7 * c * cos (43° ) ; ; ; ; c**2 -36.129c +324.48 =0 ; ; p=1; q=-36.129; r=324.48 ; ; D = q**2 - 4pr = 36.129**2 - 4 * 1 * 324.48 = 7.37545413311 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 36.13 ± sqrt{ 7.38 } }{ 2 } ; ; c_{1,2} = 18.06443643 ± 1.3578893671 ; ; c_{1} = 19.4223257971 ; ; c_{2} = 16.7065470629 ; ; ; ; text{ Factored form: } ; ; (c -19.4223257971) (c -16.7065470629) = 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 = 24.7 ; ; b = 16.9 ; ; c = 19.42 ; ;

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

p = a+b+c = 24.7+16.9+19.42 = 61.02 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61.02 }{ 2 } = 30.51 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.51 * (30.51-24.7)(30.51-16.9)(30.51-19.42) } ; ; T = sqrt{ 26761.04 } = 163.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 163.59 }{ 24.7 } = 13.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 163.59 }{ 16.9 } = 19.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 163.59 }{ 19.42 } = 16.85 ; ;

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{ 16.9**2+19.42**2-24.7**2 }{ 2 * 16.9 * 19.42 } ) = 85° 23'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.7**2+19.42**2-16.9**2 }{ 2 * 24.7 * 19.42 } ) = 43° ; ; gamma = 180° - alpha - beta = 180° - 85° 23'29" - 43° = 51° 36'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 163.59 }{ 30.51 } = 5.36 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.7 }{ 2 * sin 85° 23'29" } = 12.39 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.9**2+2 * 19.42**2 - 24.7**2 } }{ 2 } = 13.375 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.42**2+2 * 24.7**2 - 16.9**2 } }{ 2 } = 20.549 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.9**2+2 * 24.7**2 - 19.42**2 } }{ 2 } = 18.803 ; ;







#2 Obtuse scalene triangle.

Sides: a = 24.7   b = 16.9   c = 16.70765470629

Area: T = 140.7143895585
Perimeter: p = 58.30765470629
Semiperimeter: s = 29.15332735314

Angle ∠ A = α = 94.60985968388° = 94°36'31″ = 1.65112315155 rad
Angle ∠ B = β = 43° = 0.75504915784 rad
Angle ∠ C = γ = 42.39114031612° = 42°23'29″ = 0.74398695597 rad

Height: ha = 11.39438376992
Height: hb = 16.65325320219
Height: hc = 16.84553594935

Median: ma = 11.3954597728
Median: mb = 19.3188303688
Median: mc = 19.4444094767

Inradius: r = 4.82766928046
Circumradius: R = 12.39900591187

Vertex coordinates: A[16.70765470629; 0] B[0; 0] C[18.064443643; 16.84553594935]
Centroid: CG[11.5990327831; 5.61551198312]
Coordinates of the circumscribed circle: U[8.35332735314; 9.15107587812]
Coordinates of the inscribed circle: I[12.25332735314; 4.82766928046]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 85.39114031612° = 85°23'29″ = 1.65112315155 rad
∠ B' = β' = 137° = 0.75504915784 rad
∠ C' = γ' = 137.6098596839° = 137°36'31″ = 0.74398695597 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 24.7 ; ; b = 16.9 ; ; beta = 43° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.9**2 = 24.7**2 + c**2 -2 * 24.7 * c * cos (43° ) ; ; ; ; c**2 -36.129c +324.48 =0 ; ; p=1; q=-36.129; r=324.48 ; ; D = q**2 - 4pr = 36.129**2 - 4 * 1 * 324.48 = 7.37545413311 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 36.13 ± sqrt{ 7.38 } }{ 2 } ; ; c_{1,2} = 18.06443643 ± 1.3578893671 ; ; c_{1} = 19.4223257971 ; ; c_{2} = 16.7065470629 ; ; ; ; text{ Factored form: } ; ; (c -19.4223257971) (c -16.7065470629) = 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 = 24.7 ; ; b = 16.9 ; ; c = 16.71 ; ;

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

p = a+b+c = 24.7+16.9+16.71 = 58.31 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58.31 }{ 2 } = 29.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.15 * (29.15-24.7)(29.15-16.9)(29.15-16.71) } ; ; T = sqrt{ 19800.4 } = 140.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 140.71 }{ 24.7 } = 11.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 140.71 }{ 16.9 } = 16.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 140.71 }{ 16.71 } = 16.85 ; ;

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{ 16.9**2+16.71**2-24.7**2 }{ 2 * 16.9 * 16.71 } ) = 94° 36'31" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.7**2+16.71**2-16.9**2 }{ 2 * 24.7 * 16.71 } ) = 43° ; ; gamma = 180° - alpha - beta = 180° - 94° 36'31" - 43° = 42° 23'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 140.71 }{ 29.15 } = 4.83 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.7 }{ 2 * sin 94° 36'31" } = 12.39 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.9**2+2 * 16.71**2 - 24.7**2 } }{ 2 } = 11.395 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.71**2+2 * 24.7**2 - 16.9**2 } }{ 2 } = 19.318 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.9**2+2 * 24.7**2 - 16.71**2 } }{ 2 } = 19.444 ; ;
Calculate another triangle

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

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