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=75.73114136849; b=124; c=164 and a=152.1176531826; b=124; c=164.

#1 Obtuse scalene triangle.

Sides: a = 75.73114136849   b = 124   c = 164

Area: T = 4467.083283996
Perimeter: p = 363.7311413685
Semiperimeter: s = 181.8665706842

Angle ∠ A = α = 26.06109860088° = 26°3'40″ = 0.45548500122 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 107.9399013991° = 107°56'20″ = 1.88438911855 rad

Height: ha = 117.9721727255
Height: hb = 72.05497232251
Height: hc = 54.47766199995

Median: ma = 140.3644483561
Median: mb = 111.6766423247
Median: mc = 61.99001091223

Inradius: r = 24.56325352768
Circumradius: R = 86.1990142643

Vertex coordinates: A[164; 0] B[0; 0] C[52.60774604229; 54.47766199995]
Centroid: CG[72.20224868076; 18.15988733332]
Coordinates of the circumscribed circle: U[82; -26.54769525337]
Coordinates of the inscribed circle: I[57.86657068424; 24.56325352768]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.9399013991° = 153°56'20″ = 0.45548500122 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 72.06109860088° = 72°3'40″ = 1.88438911855 rad




How did we calculate this triangle?

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

b = 124 ; ; c = 164 ; ; beta = 46° ; ;

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 ; ; ; ; 124**2 = 164**2 + a**2 - 2 * 164 * a * cos 46° ; ; ; ; ; ; a**2 -227.848a +11520 =0 ; ; a=1; b=-227.848; c=11520 ; ; D = b**2 - 4ac = 227.848**2 - 4 * 1 * 11520 = 5834.68627338 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 227.85 ± sqrt{ 5834.69 } }{ 2 } ; ; a_{1,2} = 113.92397276 ± 38.1925590704 ; ; a_{1} = 152.11653183 ; ; a_{2} = 75.7314136896 ; ;
 ; ; text{ Factored form: } ; ; (a -152.11653183) (a -75.7314136896) = 0 ; ; ; ; a > 0 ; ; ; ; a = 152.117 ; ;


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 = 75.73 ; ; b = 124 ; ; c = 164 ; ;

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

p = a+b+c = 75.73+124+164 = 363.73 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 363.73 }{ 2 } = 181.87 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 181.87 * (181.87-75.73)(181.87-124)(181.87-164) } ; ; T = sqrt{ 19954829.1 } = 4467.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4467.08 }{ 75.73 } = 117.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4467.08 }{ 124 } = 72.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4467.08 }{ 164 } = 54.48 ; ;

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{ 124**2+164**2-75.73**2 }{ 2 * 124 * 164 } ) = 26° 3'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 75.73**2+164**2-124**2 }{ 2 * 75.73 * 164 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 26° 3'40" - 46° = 107° 56'20" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4467.08 }{ 181.87 } = 24.56 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 75.73 }{ 2 * sin 26° 3'40" } = 86.19 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 124**2+2 * 164**2 - 75.73**2 } }{ 2 } = 140.364 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 164**2+2 * 75.73**2 - 124**2 } }{ 2 } = 111.676 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 124**2+2 * 75.73**2 - 164**2 } }{ 2 } = 61.9 ; ;







#2 Acute scalene triangle.

Sides: a = 152.1176531826   b = 124   c = 164

Area: T = 8972.72550018
Perimeter: p = 440.1176531826
Semiperimeter: s = 220.0588265913

Angle ∠ A = α = 61.93990139912° = 61°56'20″ = 1.08110397296 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 72.06109860088° = 72°3'40″ = 1.25877014681 rad

Height: ha = 117.9721727255
Height: hb = 144.7211370997
Height: hc = 109.4233475632

Median: ma = 123.9899718266
Median: mb = 145.5121922629
Median: mc = 111.9544096072

Inradius: r = 40.77443147688
Circumradius: R = 86.1990142643

Vertex coordinates: A[164; 0] B[0; 0] C[105.6699022118; 109.4233475632]
Centroid: CG[89.89896740393; 36.47444918772]
Coordinates of the circumscribed circle: U[82; 26.54769525337]
Coordinates of the inscribed circle: I[96.05882659128; 40.77443147688]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.0610986009° = 118°3'40″ = 1.08110397296 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 107.9399013991° = 107°56'20″ = 1.25877014681 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 124 ; ; c = 164 ; ; beta = 46° ; ; : 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 ; ; ; ; 124**2 = 164**2 + a**2 - 2 * 164 * a * cos 46° ; ; ; ; ; ; a**2 -227.848a +11520 =0 ; ; a=1; b=-227.848; c=11520 ; ; D = b**2 - 4ac = 227.848**2 - 4 * 1 * 11520 = 5834.68627338 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 227.85 ± sqrt{ 5834.69 } }{ 2 } ; ; a_{1,2} = 113.92397276 ± 38.1925590704 ; ; a_{1} = 152.11653183 ; ; a_{2} = 75.7314136896 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (a -152.11653183) (a -75.7314136896) = 0 ; ; ; ; a > 0 ; ; ; ; a = 152.117 ; ; : 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 = 152.12 ; ; b = 124 ; ; c = 164 ; ;

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

p = a+b+c = 152.12+124+164 = 440.12 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 440.12 }{ 2 } = 220.06 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 220.06 * (220.06-152.12)(220.06-124)(220.06-164) } ; ; T = sqrt{ 80509793.96 } = 8972.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8972.73 }{ 152.12 } = 117.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8972.73 }{ 124 } = 144.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8972.73 }{ 164 } = 109.42 ; ;

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{ 124**2+164**2-152.12**2 }{ 2 * 124 * 164 } ) = 61° 56'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 152.12**2+164**2-124**2 }{ 2 * 152.12 * 164 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 61° 56'20" - 46° = 72° 3'40" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8972.73 }{ 220.06 } = 40.77 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 152.12 }{ 2 * sin 61° 56'20" } = 86.19 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 124**2+2 * 164**2 - 152.12**2 } }{ 2 } = 123.9 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 164**2+2 * 152.12**2 - 124**2 } }{ 2 } = 145.512 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 124**2+2 * 152.12**2 - 164**2 } }{ 2 } = 111.954 ; ;
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