Triangle calculator

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

You have entered side b, c and angle β.

Triangle has two solutions: a=3.79662124427; b=35; c=38 and a=57.68990791298; b=35; c=38.

#1 Obtuse scalene triangle.

Sides: a = 3.79662124427   b = 35   c = 38

Area: T = 42.39657960793
Perimeter: p = 76.79662124427
Semiperimeter: s = 38.39881062213

Angle ∠ A = α = 3.6555261197° = 3°39'19″ = 0.06437963429 rad
Angle ∠ B = β = 36° = 0.62883185307 rad
Angle ∠ C = γ = 140.3454738803° = 140°20'41″ = 2.449947778 rad

Height: ha = 22.33658395871
Height: hb = 2.42326169188
Height: hc = 2.23113576884

Median: ma = 36.48114636874
Median: mb = 20.56658847234
Median: mc = 16.08443282252

Inradius: r = 1.10441116412
Circumradius: R = 29.77327782923

Vertex coordinates: A[38; 0] B[0; 0] C[3.07112003804; 2.23113576884]
Centroid: CG[13.69904001268; 0.74437858961]
Coordinates of the circumscribed circle: U[19; -22.92220053059]
Coordinates of the inscribed circle: I[3.39881062213; 1.10441116412]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 176.3454738803° = 176°20'41″ = 0.06437963429 rad
∠ B' = β' = 144° = 0.62883185307 rad
∠ C' = γ' = 39.6555261197° = 39°39'19″ = 2.449947778 rad




How did we calculate this triangle?

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

b = 35 ; ; c = 38 ; ; beta = 36° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 35**2 = 38**2 + a**2 - 2 * 38 * a * cos 36° ; ; ; ; ; ; a**2 -61.485a +219 =0 ; ; p=1; q=-61.485; r=219 ; ; D = q**2 - 4pr = 61.485**2 - 4 * 1 * 219 = 2904.44107975 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 61.49 ± sqrt{ 2904.44 } }{ 2 } ; ; a_{1,2} = 30.74264579 ± 26.9464333436 ; ; a_{1} = 57.6890791336 ; ; a_{2} = 3.79621244644 ; ; ; ; text{ Factored form: } ; ; (a -57.6890791336) (a -3.79621244644) = 0 ; ; ; ; a > 0 ; ; ; ; a = 57.689 ; ;


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 = 3.8 ; ; b = 35 ; ; c = 38 ; ;

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

p = a+b+c = 3.8+35+38 = 76.8 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76.8 }{ 2 } = 38.4 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.4 * (38.4-3.8)(38.4-35)(38.4-38) } ; ; T = sqrt{ 1797.4 } = 42.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.4 }{ 3.8 } = 22.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.4 }{ 35 } = 2.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.4 }{ 38 } = 2.23 ; ;

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{ 35**2+38**2-3.8**2 }{ 2 * 35 * 38 } ) = 3° 39'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.8**2+38**2-35**2 }{ 2 * 3.8 * 38 } ) = 36° ; ; gamma = 180° - alpha - beta = 180° - 3° 39'19" - 36° = 140° 20'41" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.4 }{ 38.4 } = 1.1 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.8 }{ 2 * sin 3° 39'19" } = 29.77 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 38**2 - 3.8**2 } }{ 2 } = 36.481 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 38**2+2 * 3.8**2 - 35**2 } }{ 2 } = 20.566 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 3.8**2 - 38**2 } }{ 2 } = 16.084 ; ;







#2 Obtuse scalene triangle.

Sides: a = 57.68990791298   b = 35   c = 38

Area: T = 644.2677008686
Perimeter: p = 130.689907913
Semiperimeter: s = 65.34545395649

Angle ∠ A = α = 104.3454738803° = 104°20'41″ = 1.82111592492 rad
Angle ∠ B = β = 36° = 0.62883185307 rad
Angle ∠ C = γ = 39.6555261197° = 39°39'19″ = 0.69221148736 rad

Height: ha = 22.33658395871
Height: hb = 36.81552576392
Height: hc = 33.90987899308

Median: ma = 22.41663453152
Median: mb = 45.60444397556
Median: mc = 43.76765959999

Inradius: r = 9.86595385778
Circumradius: R = 29.77327782923

Vertex coordinates: A[38; 0] B[0; 0] C[46.67114454059; 33.90987899308]
Centroid: CG[28.22438151353; 11.30329299769]
Coordinates of the circumscribed circle: U[19; 22.92220053059]
Coordinates of the inscribed circle: I[30.34545395649; 9.86595385778]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 75.6555261197° = 75°39'19″ = 1.82111592492 rad
∠ B' = β' = 144° = 0.62883185307 rad
∠ C' = γ' = 140.3454738803° = 140°20'41″ = 0.69221148736 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 35 ; ; c = 38 ; ; beta = 36° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 35**2 = 38**2 + a**2 - 2 * 38 * a * cos 36° ; ; ; ; ; ; a**2 -61.485a +219 =0 ; ; p=1; q=-61.485; r=219 ; ; D = q**2 - 4pr = 61.485**2 - 4 * 1 * 219 = 2904.44107975 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 61.49 ± sqrt{ 2904.44 } }{ 2 } ; ; a_{1,2} = 30.74264579 ± 26.9464333436 ; ; a_{1} = 57.6890791336 ; ; a_{2} = 3.79621244644 ; ; ; ; text{ Factored form: } ; ; (a -57.6890791336) (a -3.79621244644) = 0 ; ; ; ; a > 0 ; ; ; ; a = 57.689 ; ; : 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 = 57.69 ; ; b = 35 ; ; c = 38 ; ;

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

p = a+b+c = 57.69+35+38 = 130.69 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 130.69 }{ 2 } = 65.34 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.34 * (65.34-57.69)(65.34-35)(65.34-38) } ; ; T = sqrt{ 415079.98 } = 644.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 644.27 }{ 57.69 } = 22.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 644.27 }{ 35 } = 36.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 644.27 }{ 38 } = 33.91 ; ;

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{ 35**2+38**2-57.69**2 }{ 2 * 35 * 38 } ) = 104° 20'41" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 57.69**2+38**2-35**2 }{ 2 * 57.69 * 38 } ) = 36° ; ; gamma = 180° - alpha - beta = 180° - 104° 20'41" - 36° = 39° 39'19" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 644.27 }{ 65.34 } = 9.86 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 57.69 }{ 2 * sin 104° 20'41" } = 29.77 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 38**2 - 57.69**2 } }{ 2 } = 22.416 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 38**2+2 * 57.69**2 - 35**2 } }{ 2 } = 45.604 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 57.69**2 - 38**2 } }{ 2 } = 43.767 ; ;
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