Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and angle β.

Triangle has two solutions: a=149600000; b=57910000; c=101439971.751 and a=149600000; b=57910000; c=187565035.475.

#1 Obtuse scalene triangle.

Sides: a = 149600000   b = 57910000   c = 101439971.751

Area: T = 1.96384382746E+15
Perimeter: p = 308949971.751
Semiperimeter: s = 154474985.875

Angle ∠ A = α = 138.0439960574° = 138°2'24″ = 2.40992518113 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 26.96600394264° = 26°57'36″ = 0.47105414545 rad

Height: ha = 26254596.62437
Height: hb = 67823996.80439
Height: hc = 38719329.14773

Median: ma = 35025390.56773
Median: mb = 124485830.155
Median: mc = 101461061.905

Inradius: r = 12713021.57
Circumradius: R = 111873529.201

Vertex coordinates: A[101439971.751; 0] B[0; 0] C[144502503.613; 38719329.14773]
Centroid: CG[81980825.12112; 12906443.04991]
Coordinates of the circumscribed circle: U[50719985.87553; 99715442.98797]
Coordinates of the inscribed circle: I[96564985.87553; 12713021.57]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 41.96600394264° = 41°57'36″ = 2.40992518113 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 153.0439960574° = 153°2'24″ = 0.47105414545 rad




How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

a = 149600000 ; ; b = 57910000 ; ; beta = 15° ; ;

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 57910000**2 = 149600000**2 + c**2 - 2 * 149600000 * c * cos 15° ; ; ; ; ; ; c**2 -289005007.226c +1.90265919E+16 =0 ; ; p=1; q=-289005007.226; r=1.90265919E+16 ; ; D = q**2 - 4pr = 289005007.226**2 - 4 * 1 * 1.90265919E+16 = 7.41752660152E+15 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 289005007.23 ± sqrt{ 7.41752660152E+15 } }{ 2 } = fraction{ 289005007.23 ± 4 sqrt{ 4.63595412595E+14 } }{ 2 } ; ;
c_{1,2} = 144502503.613 ± 43062531.8622 ; ; c_{1} = 187565035.475 ; ; c_{2} = 101439971.751 ; ; ; ; text{ Factored form: } ; ; (c -187565035.475) (c -101439971.751) = 0 ; ; ; ; c > 0 ; ; ; ; c = 187565035.475 ; ;
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 = 149600000 ; ; b = 57910000 ; ; c = 101439971.75 ; ;

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

p = a+b+c = 149600000+57910000+101439971.75 = 308949971.75 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 308949971.75 }{ 2 } = 154474985.88 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 154474985.88 * (154474985.88-149600000)(154474985.88-57910000)(154474985.88-101439971.75) } ; ; T = sqrt{ 3.857 * 10**{ 30 } } = 1.964 * 10**{ 15 } ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.964 * 10**{ 15 } }{ 149600000 } = 26254596.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.964 * 10**{ 15 } }{ 57910000 } = 67823996.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.964 * 10**{ 15 } }{ 101439971.75 } = 38719329.15 ; ;

7. 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{ 57910000**2+101439971.75**2-149600000**2 }{ 2 * 57910000 * 101439971.75 } ) = 138° 2'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 149600000**2+101439971.75**2-57910000**2 }{ 2 * 149600000 * 101439971.75 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 138° 2'24" - 15° = 26° 57'36" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.964 * 10**{ 15 } }{ 154474985.88 } = 12713021.57 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 149600000 }{ 2 * sin 138° 2'24" } = 111873529.2 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 57910000**2+2 * 101439971.75**2 - 149600000**2 } }{ 2 } = 35025390.567 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 101439971.75**2+2 * 149600000**2 - 57910000**2 } }{ 2 } = 124485830.155 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 57910000**2+2 * 149600000**2 - 101439971.75**2 } }{ 2 } = 101461061.905 ; ;







#2 Obtuse scalene triangle.

Sides: a = 149600000   b = 57910000   c = 187565035.475

Area: T = 3.63119617254E+15
Perimeter: p = 395075035.475
Semiperimeter: s = 197537517.738

Angle ∠ A = α = 41.96600394264° = 41°57'36″ = 0.73223408423 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 123.0439960574° = 123°2'24″ = 2.14774524235 rad

Height: ha = 48545403.37663
Height: hb = 125408260.147
Height: hc = 38719329.14773

Median: ma = 116927607.161
Median: mb = 167158634.959
Median: mc = 63809900.61775

Inradius: r = 18382311.44221
Circumradius: R = 111873529.201

Vertex coordinates: A[187565035.475; 0] B[0; 0] C[144502503.613; 38719329.14773]
Centroid: CG[110689179.696; 12906443.04991]
Coordinates of the circumscribed circle: U[93782517.73875; -60996113.83224]
Coordinates of the inscribed circle: I[139627517.738; 18382311.44221]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.0439960574° = 138°2'24″ = 0.73223408423 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 56.96600394264° = 56°57'36″ = 2.14774524235 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

a = 149600000 ; ; b = 57910000 ; ; beta = 15° ; ; : Nr. 1

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 57910000**2 = 149600000**2 + c**2 - 2 * 149600000 * c * cos 15° ; ; ; ; ; ; c**2 -289005007.226c +1.90265919E+16 =0 ; ; p=1; q=-289005007.226; r=1.90265919E+16 ; ; D = q**2 - 4pr = 289005007.226**2 - 4 * 1 * 1.90265919E+16 = 7.41752660152E+15 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 289005007.23 ± sqrt{ 7.41752660152E+15 } }{ 2 } = fraction{ 289005007.23 ± 4 sqrt{ 4.63595412595E+14 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 144502503.613 ± 43062531.8622 ; ; c_{1} = 187565035.475 ; ; c_{2} = 101439971.751 ; ; ; ; text{ Factored form: } ; ; (c -187565035.475) (c -101439971.751) = 0 ; ; ; ; c > 0 ; ; ; ; c = 187565035.475 ; ; : 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 = 149600000 ; ; b = 57910000 ; ; c = 187565035.48 ; ;

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

p = a+b+c = 149600000+57910000+187565035.48 = 395075035.48 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 395075035.48 }{ 2 } = 197537517.74 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 197537517.74 * (197537517.74-149600000)(197537517.74-57910000)(197537517.74-187565035.48) } ; ; T = sqrt{ 1.319 * 10**{ 31 } } = 3.631 * 10**{ 15 } ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.631 * 10**{ 15 } }{ 149600000 } = 48545403.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.631 * 10**{ 15 } }{ 57910000 } = 125408260.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.631 * 10**{ 15 } }{ 187565035.48 } = 38719329.15 ; ;

7. 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{ 57910000**2+187565035.48**2-149600000**2 }{ 2 * 57910000 * 187565035.48 } ) = 41° 57'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 149600000**2+187565035.48**2-57910000**2 }{ 2 * 149600000 * 187565035.48 } ) = 15° ; ; gamma = 180° - alpha - beta = 180° - 41° 57'36" - 15° = 123° 2'24" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.631 * 10**{ 15 } }{ 197537517.74 } = 18382311.44 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 149600000 }{ 2 * sin 41° 57'36" } = 111873529.2 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 57910000**2+2 * 187565035.48**2 - 149600000**2 } }{ 2 } = 116927607.161 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 187565035.48**2+2 * 149600000**2 - 57910000**2 } }{ 2 } = 167158634.959 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 57910000**2+2 * 149600000**2 - 187565035.48**2 } }{ 2 } = 63809900.618 ; ;
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