Triangle calculator SSA

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Triangle has two solutions with side c=17.39439811156 and with side c=5.34766770708

#1 Acute scalene triangle.

Sides: a = 47   b = 46   c = 17.39439811156

Area: T = 396.6176680176
Perimeter: p = 110.3943981116
Semiperimeter: s = 55.19769905578

Angle ∠ A = α = 82.47655676591° = 82°28'32″ = 1.43994702081 rad
Angle ∠ B = β = 76° = 1.32664502315 rad
Angle ∠ C = γ = 21.52444323409° = 21°31'28″ = 0.3765672214 rad

Height: ha = 16.87773055394
Height: hb = 17.24442034859
Height: hc = 45.6043899135

Median: ma = 25.63325045504
Median: mb = 26.95987701783
Median: mc = 45.68221885995

Inradius: r = 7.18554765299
Circumradius: R = 23.7044113475

Vertex coordinates: A[17.39439811156; 0] B[0; 0] C[11.37703290932; 45.6043899135]
Centroid: CG[9.58881034029; 15.20112997117]
Coordinates of the circumscribed circle: U[8.69769905578; 22.05110170032]
Coordinates of the inscribed circle: I[9.19769905578; 7.18554765299]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 97.52444323409° = 97°31'28″ = 1.43994702081 rad
∠ B' = β' = 104° = 1.32664502315 rad
∠ C' = γ' = 158.4765567659° = 158°28'32″ = 0.3765672214 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 = 47 ; ; b = 46 ; ; c = 17.39 ; ;

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

p = a+b+c = 47+46+17.39 = 110.39 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 110.39 }{ 2 } = 55.2 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 55.2 * (55.2-47)(55.2-46)(55.2-17.39) } ; ; T = sqrt{ 157304.79 } = 396.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 396.62 }{ 47 } = 16.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 396.62 }{ 46 } = 17.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 396.62 }{ 17.39 } = 45.6 ; ;

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{ 47**2-46**2-17.39**2 }{ 2 * 46 * 17.39 } ) = 82° 28'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 46**2-47**2-17.39**2 }{ 2 * 47 * 17.39 } ) = 76° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.39**2-47**2-46**2 }{ 2 * 46 * 47 } ) = 21° 31'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 396.62 }{ 55.2 } = 7.19 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 47 }{ 2 * sin 82° 28'32" } = 23.7 ; ;





#2 Obtuse scalene triangle.

Sides: a = 47   b = 46   c = 5.34766770708

Area: T = 121.9154660921
Perimeter: p = 98.34766770708
Semiperimeter: s = 49.17333385354

Angle ∠ A = α = 97.52444323409° = 97°31'28″ = 1.70221224455 rad
Angle ∠ B = β = 76° = 1.32664502315 rad
Angle ∠ C = γ = 6.47655676591° = 6°28'32″ = 0.11330199766 rad

Height: ha = 5.18878579115
Height: hb = 5.30106374313
Height: hc = 45.6043899135

Median: ma = 22.8044461797
Median: mb = 24.28656640397
Median: mc = 46.42657822882

Inradius: r = 2.47992837857
Circumradius: R = 23.7044113475

Vertex coordinates: A[5.34766770708; 0] B[0; 0] C[11.37703290932; 45.6043899135]
Centroid: CG[5.5722335388; 15.20112997117]
Coordinates of the circumscribed circle: U[2.67333385354; 23.55328821318]
Coordinates of the inscribed circle: I[3.17333385354; 2.47992837857]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 82.47655676591° = 82°28'32″ = 1.70221224455 rad
∠ B' = β' = 104° = 1.32664502315 rad
∠ C' = γ' = 173.5244432341° = 173°31'28″ = 0.11330199766 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 47 ; ; b = 46 ; ; beta = 76° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 46**2 = 47**2 + c**2 -2 * 46 * c * cos (76° ) ; ; ; ; c**2 -22.741c +93 =0 ; ; p=1; q=-22.7406581864; r=93 ; ; D = q**2 - 4pr = 22.741**2 - 4 * 1 * 93 = 145.137534749 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 22.74 ± sqrt{ 145.14 } }{ 2 } ; ; c_{1,2} = 11.3703290932 ± 6.02365202243 ; ; c_{1} = 17.3939811156 ; ;
c_{2} = 5.34667707076 ; ; ; ; (c -17.3939811156) (c -5.34667707076) = 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 = 47 ; ; b = 46 ; ; c = 5.35 ; ;

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

p = a+b+c = 47+46+5.35 = 98.35 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 98.35 }{ 2 } = 49.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 49.17 * (49.17-47)(49.17-46)(49.17-5.35) } ; ; T = sqrt{ 14863.18 } = 121.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 121.91 }{ 47 } = 5.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 121.91 }{ 46 } = 5.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 121.91 }{ 5.35 } = 45.6 ; ;

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{ 47**2-46**2-5.35**2 }{ 2 * 46 * 5.35 } ) = 97° 31'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 46**2-47**2-5.35**2 }{ 2 * 47 * 5.35 } ) = 76° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.35**2-47**2-46**2 }{ 2 * 46 * 47 } ) = 6° 28'32" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 121.91 }{ 49.17 } = 2.48 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 47 }{ 2 * sin 97° 31'28" } = 23.7 ; ;




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