Triangle calculator

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

You have entered side b, c and angle γ.

Triangle has two solutions: a=43.00549729383; b=80; c=56 and a=75.89881991381; b=80; c=56.

#1 Obtuse scalene triangle.

Sides: a = 43.00549729383   b = 80   c = 56

Area: T = 1151.038774475
Perimeter: p = 179.0054972938
Semiperimeter: s = 89.50224864692

Angle ∠ A = α = 30.92110281362° = 30°55'16″ = 0.54396737491 rad
Angle ∠ B = β = 107.0798971864° = 107°4'44″ = 1.86988806187 rad
Angle ∠ C = γ = 42° = 0.73330382858 rad

Height: ha = 53.53304485087
Height: hb = 28.77659436187
Height: hc = 41.10884908838

Median: ma = 65.61773991838
Median: mb = 29.87883173675
Median: mc = 57.79989087156

Inradius: r = 12.86603996398
Circumradius: R = 41.84553433962

Vertex coordinates: A[56; 0] B[0; 0] C[-12.63301098444; 41.10884908838]
Centroid: CG[14.45766300519; 13.70328302946]
Coordinates of the circumscribed circle: U[28; 31.09771504152]
Coordinates of the inscribed circle: I[9.50224864692; 12.86603996398]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.0798971864° = 149°4'44″ = 0.54396737491 rad
∠ B' = β' = 72.92110281362° = 72°55'16″ = 1.86988806187 rad
∠ C' = γ' = 138° = 0.73330382858 rad




How did we calculate this triangle?

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

b = 80 ; ; c = 56 ; ; gamma = 42° ; ;

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 56**2 = 80**2 + a**2 - 2 * 80 * a * cos 42° ; ; ; ; ; ; a**2 -118.903a +3264 =0 ; ; p=1; q=-118.903; r=3264 ; ; D = q**2 - 4pr = 118.903**2 - 4 * 1 * 3264 = 1081.96432983 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 118.9 ± sqrt{ 1081.96 } }{ 2 } ; ; a_{1,2} = 59.45158604 ± 16.4466130999 ; ; a_{1} = 75.8981991399 ; ; a_{2} = 43.0049729401 ; ; ; ; text{ Factored form: } ; ; (a -75.8981991399) (a -43.0049729401) = 0 ; ; ; ; a > 0 ; ; ; ; a = 75.898 ; ;


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 = 43 ; ; b = 80 ; ; c = 56 ; ;

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

p = a+b+c = 43+80+56 = 179 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 179 }{ 2 } = 89.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 89.5 * (89.5-43)(89.5-80)(89.5-56) } ; ; T = sqrt{ 1324887.89 } = 1151.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1151.04 }{ 43 } = 53.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1151.04 }{ 80 } = 28.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1151.04 }{ 56 } = 41.11 ; ;

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{ 80**2+56**2-43**2 }{ 2 * 80 * 56 } ) = 30° 55'16" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 43**2+56**2-80**2 }{ 2 * 43 * 56 } ) = 107° 4'44" ; ; gamma = 180° - alpha - beta = 180° - 30° 55'16" - 107° 4'44" = 42° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1151.04 }{ 89.5 } = 12.86 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 43 }{ 2 * sin 30° 55'16" } = 41.85 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 56**2 - 43**2 } }{ 2 } = 65.617 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 43**2 - 80**2 } }{ 2 } = 29.878 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 43**2 - 56**2 } }{ 2 } = 57.799 ; ;







#2 Acute scalene triangle.

Sides: a = 75.89881991381   b = 80   c = 56

Area: T = 2031.432232043
Perimeter: p = 211.8988199138
Semiperimeter: s = 105.9499099569

Angle ∠ A = α = 65.07989718638° = 65°4'44″ = 1.13658423328 rad
Angle ∠ B = β = 72.92110281362° = 72°55'16″ = 1.27327120349 rad
Angle ∠ C = γ = 42° = 0.73330382858 rad

Height: ha = 53.53304485087
Height: hb = 50.78658080108
Height: hc = 72.55111543011

Median: ma = 57.68876576219
Median: mb = 53.3699170091
Median: mc = 72.77554650703

Inradius: r = 19.1743662907
Circumradius: R = 41.84553433962

Vertex coordinates: A[56; 0] B[0; 0] C[22.29105056464; 72.55111543011]
Centroid: CG[26.09768352155; 24.18437181004]
Coordinates of the circumscribed circle: U[28; 31.09771504152]
Coordinates of the inscribed circle: I[25.9499099569; 19.1743662907]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114.9211028136° = 114°55'16″ = 1.13658423328 rad
∠ B' = β' = 107.0798971864° = 107°4'44″ = 1.27327120349 rad
∠ C' = γ' = 138° = 0.73330382858 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 80 ; ; c = 56 ; ; gamma = 42° ; ; : Nr. 1

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 56**2 = 80**2 + a**2 - 2 * 80 * a * cos 42° ; ; ; ; ; ; a**2 -118.903a +3264 =0 ; ; p=1; q=-118.903; r=3264 ; ; D = q**2 - 4pr = 118.903**2 - 4 * 1 * 3264 = 1081.96432983 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 118.9 ± sqrt{ 1081.96 } }{ 2 } ; ; a_{1,2} = 59.45158604 ± 16.4466130999 ; ; a_{1} = 75.8981991399 ; ; a_{2} = 43.0049729401 ; ; ; ; text{ Factored form: } ; ; (a -75.8981991399) (a -43.0049729401) = 0 ; ; ; ; a > 0 ; ; ; ; a = 75.898 ; ; : 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 = 75.9 ; ; b = 80 ; ; c = 56 ; ;

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

p = a+b+c = 75.9+80+56 = 211.9 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 211.9 }{ 2 } = 105.95 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 105.95 * (105.95-75.9)(105.95-80)(105.95-56) } ; ; T = sqrt{ 4126717.27 } = 2031.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2031.43 }{ 75.9 } = 53.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2031.43 }{ 80 } = 50.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2031.43 }{ 56 } = 72.55 ; ;

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{ 80**2+56**2-75.9**2 }{ 2 * 80 * 56 } ) = 65° 4'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 75.9**2+56**2-80**2 }{ 2 * 75.9 * 56 } ) = 72° 55'16" ; ; gamma = 180° - alpha - beta = 180° - 65° 4'44" - 72° 55'16" = 42° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2031.43 }{ 105.95 } = 19.17 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 75.9 }{ 2 * sin 65° 4'44" } = 41.85 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 56**2 - 75.9**2 } }{ 2 } = 57.688 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 75.9**2 - 80**2 } }{ 2 } = 53.369 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 75.9**2 - 56**2 } }{ 2 } = 72.775 ; ;
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