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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 43**2-80**2-56**2 }{ 2 * 80 * 56 } ) = 30° 55'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 80**2-43**2-56**2 }{ 2 * 43 * 56 } ) = 107° 4'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 56**2-43**2-80**2 }{ 2 * 80 * 43 } ) = 42° ; ;

6. Inradius

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

7. Circumradius

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





#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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 75.9**2-80**2-56**2 }{ 2 * 80 * 56 } ) = 65° 4'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 80**2-75.9**2-56**2 }{ 2 * 75.9 * 56 } ) = 72° 55'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 56**2-75.9**2-80**2 }{ 2 * 80 * 75.9 } ) = 42° ; ;

6. Inradius

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

7. Circumradius

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




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