Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=48.13989910986 and with side c=5.48441199197

#1 Obtuse scalene triangle.

Sides: a = 35   b = 31   c = 48.13989910986

Area: T = 541.5055072867
Perimeter: p = 114.1398991099
Semiperimeter: s = 57.06994955493

Angle ∠ A = α = 46.52994381208° = 46°31'46″ = 0.81220918943 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 93.47105618792° = 93°28'14″ = 1.63113690585 rad

Height: ha = 30.9433147021
Height: hb = 34.93658111527
Height: hc = 22.4987566339

Median: ma = 36.50993307525
Median: mb = 39.12771163261
Median: mc = 22.66440548888

Inradius: r = 9.48985204023
Circumradius: R = 24.11437193163

Vertex coordinates: A[48.13989910986; 0] B[0; 0] C[26.81215555092; 22.4987566339]
Centroid: CG[24.98435155359; 7.49991887797]
Coordinates of the circumscribed circle: U[24.06994955493; -1.46597408227]
Coordinates of the inscribed circle: I[26.06994955493; 9.48985204023]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.4710561879° = 133°28'14″ = 0.81220918943 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 86.52994381208° = 86°31'46″ = 1.63113690585 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 31 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 31**2 = 35**2 + c**2 -2 * 35 * c * cos (40° ) ; ; ; ; c**2 -53.623c +264 =0 ; ; p=1; q=-53.623; r=264 ; ; D = q**2 - 4pr = 53.623**2 - 4 * 1 * 264 = 1819.43803528 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 53.62 ± sqrt{ 1819.44 } }{ 2 } ; ;
c_{1,2} = 26.81155551 ± 21.3274355894 ; ; c_{1} = 48.1389910986 ; ; c_{2} = 5.48411991974 ; ; ; ; text{ Factored form: } ; ; (c -48.1389910986) (c -5.48411991974) = 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 = 35 ; ; b = 31 ; ; c = 48.14 ; ;

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

p = a+b+c = 35+31+48.14 = 114.14 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 114.14 }{ 2 } = 57.07 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.07 * (57.07-35)(57.07-31)(57.07-48.14) } ; ; T = sqrt{ 293227.74 } = 541.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 541.51 }{ 35 } = 30.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 541.51 }{ 31 } = 34.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 541.51 }{ 48.14 } = 22.5 ; ;

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{ 31**2+48.14**2-35**2 }{ 2 * 31 * 48.14 } ) = 46° 31'46" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+48.14**2-31**2 }{ 2 * 35 * 48.14 } ) = 40° ; ;
 gamma = 180° - alpha - beta = 180° - 46° 31'46" - 40° = 93° 28'14" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 541.51 }{ 57.07 } = 9.49 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 46° 31'46" } = 24.11 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 48.14**2 - 35**2 } }{ 2 } = 36.509 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.14**2+2 * 35**2 - 31**2 } }{ 2 } = 39.127 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 35**2 - 48.14**2 } }{ 2 } = 22.664 ; ;



#2 Obtuse scalene triangle.

Sides: a = 35   b = 31   c = 5.48441199197

Area: T = 61.69896758528
Perimeter: p = 71.48441199197
Semiperimeter: s = 35.74220599599

Angle ∠ A = α = 133.4710561879° = 133°28'14″ = 2.33295007593 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 6.52994381208° = 6°31'46″ = 0.11439601935 rad

Height: ha = 3.52551243344
Height: hb = 3.98799790873
Height: hc = 22.4987566339

Median: ma = 13.75881897664
Median: mb = 19.68796286969
Median: mc = 32.94766403018

Inradius: r = 1.72659686745
Circumradius: R = 24.11437193163

Vertex coordinates: A[5.48441199197; 0] B[0; 0] C[26.81215555092; 22.4987566339]
Centroid: CG[10.7655225143; 7.49991887797]
Coordinates of the circumscribed circle: U[2.74220599599; 23.95773071618]
Coordinates of the inscribed circle: I[4.74220599599; 1.72659686745]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 46.52994381208° = 46°31'46″ = 2.33295007593 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 173.4710561879° = 173°28'14″ = 0.11439601935 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 35 ; ; b = 31 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 31**2 = 35**2 + c**2 -2 * 35 * c * cos (40° ) ; ; ; ; c**2 -53.623c +264 =0 ; ; p=1; q=-53.623; r=264 ; ; D = q**2 - 4pr = 53.623**2 - 4 * 1 * 264 = 1819.43803528 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 53.62 ± sqrt{ 1819.44 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 26.81155551 ± 21.3274355894 ; ; c_{1} = 48.1389910986 ; ; c_{2} = 5.48411991974 ; ; ; ; text{ Factored form: } ; ; (c -48.1389910986) (c -5.48411991974) = 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 = 35 ; ; b = 31 ; ; c = 5.48 ; ;

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

p = a+b+c = 35+31+5.48 = 71.48 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71.48 }{ 2 } = 35.74 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.74 * (35.74-35)(35.74-31)(35.74-5.48) } ; ; T = sqrt{ 3805.62 } = 61.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 61.69 }{ 35 } = 3.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 61.69 }{ 31 } = 3.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 61.69 }{ 5.48 } = 22.5 ; ;

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{ 31**2+5.48**2-35**2 }{ 2 * 31 * 5.48 } ) = 133° 28'14" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+5.48**2-31**2 }{ 2 * 35 * 5.48 } ) = 40° ; ;
 gamma = 180° - alpha - beta = 180° - 133° 28'14" - 40° = 6° 31'46" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 61.69 }{ 35.74 } = 1.73 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 133° 28'14" } = 24.11 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 5.48**2 - 35**2 } }{ 2 } = 13.758 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.48**2+2 * 35**2 - 31**2 } }{ 2 } = 19.68 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 35**2 - 5.48**2 } }{ 2 } = 32.947 ; ;
Calculate another triangle


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

See more information about triangles or more details on solving triangles.