Triangle calculator SSA

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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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 31**2 = 35**2 + c**2 -2 * 31 * c * cos (40° ) ; ; ; ; c**2 -53.623c +264 =0 ; ; p=1; q=-53.6231110183; 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} = 26.8115555092 ± 21.3274355894 ; ; c_{1} = 48.1389910986 ; ;
c_{2} = 5.48411991974 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 35**2-31**2-5.48**2 }{ 2 * 31 * 5.48 } ) = 133° 28'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-35**2-5.48**2 }{ 2 * 35 * 5.48 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.48**2-35**2-31**2 }{ 2 * 31 * 35 } ) = 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 ; ;




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