Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=45.23333411107 and with side c=11.40875146193

#1 Acute scalene triangle.

Sides: a = 46   b = 40   c = 45.23333411107

Area: T = 819.8220261985
Perimeter: p = 131.2333341111
Semiperimeter: s = 65.61766705553

Angle ∠ A = α = 64.98770666671° = 64°59'13″ = 1.13442382846 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 63.01329333329° = 63°47″ = 1.1099783158 rad

Height: ha = 35.64443592168
Height: hb = 40.99110130993
Height: hc = 36.24884946659

Median: ma = 35.97325947635
Median: mb = 411.0003362671
Median: mc = 36.69444984022

Inradius: r = 12.49440850404
Circumradius: R = 25.38803643015

Vertex coordinates: A[45.23333411107; 0] B[0; 0] C[28.3220427865; 36.24884946659]
Centroid: CG[24.51879229919; 12.08328315553]
Coordinates of the circumscribed circle: U[22.61766705553; 11.51773393224]
Coordinates of the inscribed circle: I[25.61766705553; 12.49440850404]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.0132933333° = 115°47″ = 1.13442382846 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 116.9877066667° = 116°59'13″ = 1.1099783158 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 46 ; ; b = 40 ; ; beta = 52° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 40**2 = 46**2 + c**2 -2 * 46 * c * cos (52° ) ; ; ; ; c**2 -56.641c +516 =0 ; ; p=1; q=-56.641; r=516 ; ; D = q**2 - 4pr = 56.641**2 - 4 * 1 * 516 = 1144.18653782 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 56.64 ± sqrt{ 1144.19 } }{ 2 } ; ; c_{1,2} = 28.32042786 ± 16.9129132457 ; ; c_{1} = 45.2333411057 ; ;
c_{2} = 11.4075146143 ; ; ; ; text{ Factored form: } ; ; (c -45.2333411057) (c -11.4075146143) = 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 = 46 ; ; b = 40 ; ; c = 45.23 ; ;

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

p = a+b+c = 46+40+45.23 = 131.23 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 131.23 }{ 2 } = 65.62 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.62 * (65.62-46)(65.62-40)(65.62-45.23) } ; ; T = sqrt{ 672105.26 } = 819.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 819.82 }{ 46 } = 35.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 819.82 }{ 40 } = 40.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 819.82 }{ 45.23 } = 36.25 ; ;

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{ 40**2+45.23**2-46**2 }{ 2 * 40 * 45.23 } ) = 64° 59'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+45.23**2-40**2 }{ 2 * 46 * 45.23 } ) = 52° ; ; gamma = 180° - alpha - beta = 180° - 64° 59'13" - 52° = 63° 47" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 819.82 }{ 65.62 } = 12.49 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46 }{ 2 * sin 64° 59'13" } = 25.38 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 45.23**2 - 46**2 } }{ 2 } = 35.973 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.23**2+2 * 46**2 - 40**2 } }{ 2 } = 41 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 46**2 - 45.23**2 } }{ 2 } = 36.694 ; ;







#2 Obtuse scalene triangle.

Sides: a = 46   b = 40   c = 11.40875146193

Area: T = 206.7532616415
Perimeter: p = 97.40875146193
Semiperimeter: s = 48.70437573097

Angle ∠ A = α = 115.0132933333° = 115°47″ = 2.0077354369 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 12.98770666671° = 12°59'13″ = 0.22766670735 rad

Height: ha = 8.98992441919
Height: hb = 10.33876308207
Height: hc = 36.24884946659

Median: ma = 18.33220946674
Median: mb = 26.89898809015
Median: mc = 42.72554859838

Inradius: r = 4.24551060829
Circumradius: R = 25.38803643015

Vertex coordinates: A[11.40875146193; 0] B[0; 0] C[28.3220427865; 36.24884946659]
Centroid: CG[13.24326474948; 12.08328315553]
Coordinates of the circumscribed circle: U[5.70437573097; 24.73111553436]
Coordinates of the inscribed circle: I[8.70437573097; 4.24551060829]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.98770666671° = 64°59'13″ = 2.0077354369 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 167.0132933333° = 167°47″ = 0.22766670735 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 46 ; ; b = 40 ; ; beta = 52° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 40**2 = 46**2 + c**2 -2 * 46 * c * cos (52° ) ; ; ; ; c**2 -56.641c +516 =0 ; ; p=1; q=-56.641; r=516 ; ; D = q**2 - 4pr = 56.641**2 - 4 * 1 * 516 = 1144.18653782 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 56.64 ± sqrt{ 1144.19 } }{ 2 } ; ; c_{1,2} = 28.32042786 ± 16.9129132457 ; ; c_{1} = 45.2333411057 ; ; : Nr. 1
c_{2} = 11.4075146143 ; ; ; ; text{ Factored form: } ; ; (c -45.2333411057) (c -11.4075146143) = 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 = 46 ; ; b = 40 ; ; c = 11.41 ; ;

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

p = a+b+c = 46+40+11.41 = 97.41 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 97.41 }{ 2 } = 48.7 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 48.7 * (48.7-46)(48.7-40)(48.7-11.41) } ; ; T = sqrt{ 42746.64 } = 206.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 206.75 }{ 46 } = 8.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 206.75 }{ 40 } = 10.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 206.75 }{ 11.41 } = 36.25 ; ;

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{ 40**2+11.41**2-46**2 }{ 2 * 40 * 11.41 } ) = 115° 47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+11.41**2-40**2 }{ 2 * 46 * 11.41 } ) = 52° ; ; gamma = 180° - alpha - beta = 180° - 115° 47" - 52° = 12° 59'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 206.75 }{ 48.7 } = 4.25 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46 }{ 2 * sin 115° 47" } = 25.38 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 11.41**2 - 46**2 } }{ 2 } = 18.332 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.41**2+2 * 46**2 - 40**2 } }{ 2 } = 26.89 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 46**2 - 11.41**2 } }{ 2 } = 42.725 ; ;
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.