Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=59; b=4.35664463888; c=62 and a=59; b=83.32547944784; c=62.

#1 Obtuse scalene triangle.

Sides: a = 59   b = 4.35664463888   c = 62

Area: T = 95.49546562845
Perimeter: p = 125.3566446389
Semiperimeter: s = 62.67882231944

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 2.99328538608° = 2°59'34″ = 0.05222351539 rad
Angle ∠ C = γ = 132.0077146139° = 132°26″ = 2.30439593363 rad

Height: ha = 3.23771069927
Height: hb = 43.84106204336
Height: hc = 3.08804727834

Median: ma = 32.57766682239
Median: mb = 60.47993794918
Median: mc = 28.08989535684

Inradius: r = 1.52435699325
Circumradius: R = 41.719930009

Vertex coordinates: A[62; 0] B[0; 0] C[58.92195272166; 3.08804727834]
Centroid: CG[40.30765090722; 1.02768242611]
Coordinates of the circumscribed circle: U[31; -27.92195272166]
Coordinates of the inscribed circle: I[58.32217768056; 1.52435699325]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 177.0077146139° = 177°26″ = 0.05222351539 rad
∠ C' = γ' = 47.99328538608° = 47°59'34″ = 2.30439593363 rad




How did we calculate this triangle?

1. Input data entered: side a, c and angle α.

a = 59 ; ; c = 62 ; ; alpha = 45° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 59**2 = 62**2 + b**2 - 2 * 62 * b * cos 45° ; ; ; ; ; ; b**2 -87.681b +363 =0 ; ; p=1; q=-87.681; r=363 ; ; D = q**2 - 4pr = 87.681**2 - 4 * 1 * 363 = 6236 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 87.68 ± sqrt{ 6236 } }{ 2 } ; ; b_{1,2} = 43.84062043 ± 39.4841740448 ; ; b_{1} = 83.3247944748 ; ; b_{2} = 4.3564463852 ; ; ; ; text{ Factored form: } ; ; (b -83.3247944748) (b -4.3564463852) = 0 ; ; ; ; b > 0 ; ; ; ; b = 83.325 ; ;


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 = 59 ; ; b = 4.36 ; ; c = 62 ; ;

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

p = a+b+c = 59+4.36+62 = 125.36 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 125.36 }{ 2 } = 62.68 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.68 * (62.68-59)(62.68-4.36)(62.68-62) } ; ; T = sqrt{ 9119.23 } = 95.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 95.49 }{ 59 } = 3.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 95.49 }{ 4.36 } = 43.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 95.49 }{ 62 } = 3.08 ; ;

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{ 4.36**2+62**2-59**2 }{ 2 * 4.36 * 62 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 59**2+62**2-4.36**2 }{ 2 * 59 * 62 } ) = 2° 59'34" ; ; gamma = 180° - alpha - beta = 180° - 45° - 2° 59'34" = 132° 26" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 95.49 }{ 62.68 } = 1.52 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 59 }{ 2 * sin 45° } = 41.72 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.36**2+2 * 62**2 - 59**2 } }{ 2 } = 32.577 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 59**2 - 4.36**2 } }{ 2 } = 60.479 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.36**2+2 * 59**2 - 62**2 } }{ 2 } = 28.089 ; ;







#2 Acute scalene triangle.

Sides: a = 59   b = 83.32547944784   c = 62

Area: T = 1826.505534372
Perimeter: p = 204.3254794478
Semiperimeter: s = 102.1622397239

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 87.00771461392° = 87°26″ = 1.51985611729 rad
Angle ∠ C = γ = 47.99328538608° = 47°59'34″ = 0.83876333173 rad

Height: ha = 61.91554353802
Height: hb = 43.84106204336
Height: hc = 58.92195272166

Median: ma = 67.25551907843
Median: mb = 43.89546996377
Median: mc = 65.219977521

Inradius: r = 17.8788450321
Circumradius: R = 41.719930009

Vertex coordinates: A[62; 0] B[0; 0] C[3.08804727834; 58.92195272166]
Centroid: CG[21.69334909278; 19.64398424055]
Coordinates of the circumscribed circle: U[31; 27.92195272166]
Coordinates of the inscribed circle: I[18.83876027608; 17.8788450321]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 92.99328538608° = 92°59'34″ = 1.51985611729 rad
∠ C' = γ' = 132.0077146139° = 132°26″ = 0.83876333173 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side a, c and angle α.

a = 59 ; ; c = 62 ; ; alpha = 45° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 59**2 = 62**2 + b**2 - 2 * 62 * b * cos 45° ; ; ; ; ; ; b**2 -87.681b +363 =0 ; ; p=1; q=-87.681; r=363 ; ; D = q**2 - 4pr = 87.681**2 - 4 * 1 * 363 = 6236 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 87.68 ± sqrt{ 6236 } }{ 2 } ; ; b_{1,2} = 43.84062043 ± 39.4841740448 ; ; b_{1} = 83.3247944748 ; ; b_{2} = 4.3564463852 ; ; ; ; text{ Factored form: } ; ; (b -83.3247944748) (b -4.3564463852) = 0 ; ; ; ; b > 0 ; ; ; ; b = 83.325 ; ; : 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 = 59 ; ; b = 83.32 ; ; c = 62 ; ;

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

p = a+b+c = 59+83.32+62 = 204.32 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 204.32 }{ 2 } = 102.16 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.16 * (102.16-59)(102.16-83.32)(102.16-62) } ; ; T = sqrt{ 3336121.77 } = 1826.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1826.51 }{ 59 } = 61.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1826.51 }{ 83.32 } = 43.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1826.51 }{ 62 } = 58.92 ; ;

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{ 83.32**2+62**2-59**2 }{ 2 * 83.32 * 62 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 59**2+62**2-83.32**2 }{ 2 * 59 * 62 } ) = 87° 26" ; ; gamma = 180° - alpha - beta = 180° - 45° - 87° 26" = 47° 59'34" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1826.51 }{ 102.16 } = 17.88 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 59 }{ 2 * sin 45° } = 41.72 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 83.32**2+2 * 62**2 - 59**2 } }{ 2 } = 67.255 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 59**2 - 83.32**2 } }{ 2 } = 43.895 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 83.32**2+2 * 59**2 - 62**2 } }{ 2 } = 65.2 ; ;
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