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?

1. Use Law of Cosines

a = 47 ; ; b = 46 ; ; beta = 76° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 46**2 = 47**2 + c**2 -2 * 47 * c * cos (76° ) ; ; ; ; c**2 -22.741c +93 =0 ; ; p=1; q=-22.741; 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} = 11.37032909 ± 6.02365202243 ; ; c_{1} = 17.3939811124 ; ;
c_{2} = 5.34667706757 ; ; ; ; text{ Factored form: } ; ; (c -17.3939811124) (c -5.34667706757) = 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 = 17.39 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 46**2+17.39**2-47**2 }{ 2 * 46 * 17.39 } ) = 82° 28'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 47**2+17.39**2-46**2 }{ 2 * 47 * 17.39 } ) = 76° ; ; gamma = 180° - alpha - beta = 180° - 82° 28'32" - 76° = 21° 31'28" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 17.39**2 - 47**2 } }{ 2 } = 25.633 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.39**2+2 * 47**2 - 46**2 } }{ 2 } = 26.959 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 47**2 - 17.39**2 } }{ 2 } = 45.682 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 47 ; ; b = 46 ; ; beta = 76° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 46**2 = 47**2 + c**2 -2 * 47 * c * cos (76° ) ; ; ; ; c**2 -22.741c +93 =0 ; ; p=1; q=-22.741; 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} = 11.37032909 ± 6.02365202243 ; ; c_{1} = 17.3939811124 ; ; : Nr. 1
c_{2} = 5.34667706757 ; ; ; ; text{ Factored form: } ; ; (c -17.3939811124) (c -5.34667706757) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 46**2+5.35**2-47**2 }{ 2 * 46 * 5.35 } ) = 97° 31'28" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 47**2+5.35**2-46**2 }{ 2 * 47 * 5.35 } ) = 76° ; ; gamma = 180° - alpha - beta = 180° - 97° 31'28" - 76° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 5.35**2 - 47**2 } }{ 2 } = 22.804 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.35**2+2 * 47**2 - 46**2 } }{ 2 } = 24.286 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 46**2+2 * 47**2 - 5.35**2 } }{ 2 } = 46.426 ; ;
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