Triangle calculator SSA

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Triangle has two solutions with side c=8605.469875945 and with side c=1373.65334674

#1 Acute scalene triangle.

Sides: a = 7986   b = 7208   c = 8605.469875945

Area: T = 26829360.60114
Perimeter: p = 23799.46987595
Semiperimeter: s = 11899.73443797

Angle ∠ A = α = 59.89106439657° = 59°53'26″ = 1.04552889283 rad
Angle ∠ B = β = 51.3333333° = 51°20' = 0.89659356769 rad
Angle ∠ C = γ = 68.77660230343° = 68°46'34″ = 1.22003680484 rad

Height: ha = 6719.099857285
Height: hb = 7444.329869073
Height: hc = 6235.421106801

Median: ma = 6860.075502036
Median: mb = 7478.391075503
Median: mc = 6273.134373503

Inradius: r = 2254.618844317
Circumradius: R = 4615.814402219

Vertex coordinates: A[8605.469875945; 0] B[0; 0] C[4989.561111342; 6235.421106801]
Centroid: CG[4531.677662429; 2078.474368934]
Coordinates of the circumscribed circle: U[4302.734437973; 1670.993250297]
Coordinates of the inscribed circle: I[4691.734437973; 2254.618844317]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.1099356034° = 120°6'34″ = 1.04552889283 rad
∠ B' = β' = 128.6676667° = 128°40' = 0.89659356769 rad
∠ C' = γ' = 111.2243976966° = 111°13'26″ = 1.22003680484 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 7986 ; ; b = 7208 ; ; beta = 51° 20' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7208**2 = 7986**2 + c**2 -2 * 7986 * c * cos (51° 20') ; ; ; ; c**2 -9979.122c +11820932 =0 ; ; p=1; q=-9979.122; r=11820932 ; ; D = q**2 - 4pr = 9979.122**2 - 4 * 1 * 11820932 = 52299152.4184 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9979.12 ± sqrt{ 52299152.42 } }{ 2 } ; ; c_{1,2} = 4989.56111342 ± 3615.90764603 ; ;
c_{1} = 8605.46875945 ; ; c_{2} = 1373.65346739 ; ; ; ; text{ Factored form: } ; ; (c -8605.46875945) (c -1373.65346739) = 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 = 7986 ; ; b = 7208 ; ; c = 8605.47 ; ;

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

p = a+b+c = 7986+7208+8605.47 = 23799.47 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23799.47 }{ 2 } = 11899.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11899.73 * (11899.73-7986)(11899.73-7208)(11899.73-8605.47) } ; ; T = sqrt{ 7.198 * 10**{ 14 } } = 26829360.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26829360.6 }{ 7986 } = 6719.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26829360.6 }{ 7208 } = 7444.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26829360.6 }{ 8605.47 } = 6235.42 ; ;

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{ 7208**2+8605.47**2-7986**2 }{ 2 * 7208 * 8605.47 } ) = 59° 53'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7986**2+8605.47**2-7208**2 }{ 2 * 7986 * 8605.47 } ) = 51° 20' ; ; gamma = 180° - alpha - beta = 180° - 59° 53'26" - 51° 20' = 68° 46'34" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26829360.6 }{ 11899.73 } = 2254.62 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7986 }{ 2 * sin 59° 53'26" } = 4615.81 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7208**2+2 * 8605.47**2 - 7986**2 } }{ 2 } = 6860.075 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8605.47**2+2 * 7986**2 - 7208**2 } }{ 2 } = 7478.391 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7208**2+2 * 7986**2 - 8605.47**2 } }{ 2 } = 6273.134 ; ;







#2 Obtuse scalene triangle.

Sides: a = 7986   b = 7208   c = 1373.65334674

Area: T = 4282653.885538
Perimeter: p = 16567.65334674
Semiperimeter: s = 8283.82767337

Angle ∠ A = α = 120.1099356034° = 120°6'34″ = 2.09663037252 rad
Angle ∠ B = β = 51.3333333° = 51°20' = 0.89659356769 rad
Angle ∠ C = γ = 8.55773109657° = 8°33'26″ = 0.14993532515 rad

Height: ha = 1072.544041707
Height: hb = 1188.306573956
Height: hc = 6235.421106801

Median: ma = 3313.162237517
Median: mb = 4454.51994942
Median: mc = 7575.883272335

Inradius: r = 516.9989795061
Circumradius: R = 4615.814402219

Vertex coordinates: A[1373.65334674; 0] B[0; 0] C[4989.561111342; 6235.421106801]
Centroid: CG[2121.072152694; 2078.474368934]
Coordinates of the circumscribed circle: U[686.8276733699; 4564.429856504]
Coordinates of the inscribed circle: I[1075.82767337; 516.9989795061]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 59.89106439657° = 59°53'26″ = 2.09663037252 rad
∠ B' = β' = 128.6676667° = 128°40' = 0.89659356769 rad
∠ C' = γ' = 171.4432689034° = 171°26'34″ = 0.14993532515 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 7986 ; ; b = 7208 ; ; beta = 51° 20' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 7208**2 = 7986**2 + c**2 -2 * 7986 * c * cos (51° 20') ; ; ; ; c**2 -9979.122c +11820932 =0 ; ; p=1; q=-9979.122; r=11820932 ; ; D = q**2 - 4pr = 9979.122**2 - 4 * 1 * 11820932 = 52299152.4184 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 9979.12 ± sqrt{ 52299152.42 } }{ 2 } ; ; c_{1,2} = 4989.56111342 ± 3615.90764603 ; ; : Nr. 1
c_{1} = 8605.46875945 ; ; c_{2} = 1373.65346739 ; ; ; ; text{ Factored form: } ; ; (c -8605.46875945) (c -1373.65346739) = 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 = 7986 ; ; b = 7208 ; ; c = 1373.65 ; ;

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

p = a+b+c = 7986+7208+1373.65 = 16567.65 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16567.65 }{ 2 } = 8283.83 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8283.83 * (8283.83-7986)(8283.83-7208)(8283.83-1373.65) } ; ; T = sqrt{ 1.834 * 10**{ 13 } } = 4282653.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4282653.89 }{ 7986 } = 1072.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4282653.89 }{ 7208 } = 1188.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4282653.89 }{ 1373.65 } = 6235.42 ; ;

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{ 7208**2+1373.65**2-7986**2 }{ 2 * 7208 * 1373.65 } ) = 120° 6'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7986**2+1373.65**2-7208**2 }{ 2 * 7986 * 1373.65 } ) = 51° 20' ; ; gamma = 180° - alpha - beta = 180° - 120° 6'34" - 51° 20' = 8° 33'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4282653.89 }{ 8283.83 } = 516.99 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7986 }{ 2 * sin 120° 6'34" } = 4615.81 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7208**2+2 * 1373.65**2 - 7986**2 } }{ 2 } = 3313.162 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1373.65**2+2 * 7986**2 - 7208**2 } }{ 2 } = 4454.519 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7208**2+2 * 7986**2 - 1373.65**2 } }{ 2 } = 7575.883 ; ;
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