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=14.79221109117; b=33; c=42 and a=45.63224323167; b=33; c=42.

#1 Obtuse scalene triangle.

Sides: a = 14.79221109117   b = 33   c = 42

Area: T = 215.7854736893
Perimeter: p = 89.79221109117
Semiperimeter: s = 44.89660554559

Angle ∠ A = α = 18.1422275913° = 18°8'32″ = 0.31766424485 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 117.8587724087° = 117°51'28″ = 2.05770053342 rad

Height: ha = 29.17656515593
Height: hb = 13.0787862842
Height: hc = 10.27554636616

Median: ma = 37.03877964206
Median: mb = 26.81770332552
Median: mc = 14.59112053173

Inradius: r = 4.80663183882
Circumradius: R = 23.75326829038

Vertex coordinates: A[42; 0] B[0; 0] C[10.64105541098; 10.27554636616]
Centroid: CG[17.54768513699; 3.42551545539]
Coordinates of the circumscribed circle: U[21; -11.09990965907]
Coordinates of the inscribed circle: I[11.89660554559; 4.80663183882]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.8587724087° = 161°51'28″ = 0.31766424485 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 62.1422275913° = 62°8'32″ = 2.05770053342 rad


How did we calculate this triangle?

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

b = 33 ; ; c = 42 ; ; beta = 44° ; ;

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 ; ; ; ; 33**2 = 42**2 + a**2 - 2 * 42 * a * cos 44° ; ; ; ; ; ; a**2 -60.425a +675 =0 ; ; p=1; q=-60.425; r=675 ; ; D = q**2 - 4pr = 60.425**2 - 4 * 1 * 675 = 951.125424366 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.42 ± sqrt{ 951.13 } }{ 2 } ; ;
a_{1,2} = 30.21227161 ± 15.4201607025 ; ; a_{1} = 45.6324323167 ; ; a_{2} = 14.7921109117 ; ; ; ; text{ Factored form: } ; ; (a -45.6324323167) (a -14.7921109117) = 0 ; ; ; ; a > 0 ; ; ; ; a = 45.632 ; ;
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 = 14.79 ; ; b = 33 ; ; c = 42 ; ;

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

p = a+b+c = 14.79+33+42 = 89.79 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 89.79 }{ 2 } = 44.9 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 44.9 * (44.9-14.79)(44.9-33)(44.9-42) } ; ; T = sqrt{ 46563.05 } = 215.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 215.78 }{ 14.79 } = 29.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 215.78 }{ 33 } = 13.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 215.78 }{ 42 } = 10.28 ; ;

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{ 33**2+42**2-14.79**2 }{ 2 * 33 * 42 } ) = 18° 8'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14.79**2+42**2-33**2 }{ 2 * 14.79 * 42 } ) = 44° ; ;
 gamma = 180° - alpha - beta = 180° - 18° 8'32" - 44° = 117° 51'28" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 215.78 }{ 44.9 } = 4.81 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 14.79 }{ 2 * sin 18° 8'32" } = 23.75 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 42**2 - 14.79**2 } }{ 2 } = 37.038 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 14.79**2 - 33**2 } }{ 2 } = 26.817 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 14.79**2 - 42**2 } }{ 2 } = 14.591 ; ;



#2 Acute scalene triangle.

Sides: a = 45.63224323167   b = 33   c = 42

Area: T = 665.6787972538
Perimeter: p = 120.6322432317
Semiperimeter: s = 60.31662161584

Angle ∠ A = α = 73.8587724087° = 73°51'28″ = 1.28990604633 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 62.1422275913° = 62°8'32″ = 1.08545873194 rad

Height: ha = 29.17656515593
Height: hb = 40.34441195477
Height: hc = 31.69989510732

Median: ma = 30.09985096012
Median: mb = 40.63113849084
Median: mc = 33.8332816016

Inradius: r = 11.03664677186
Circumradius: R = 23.75326829038

Vertex coordinates: A[42; 0] B[0; 0] C[32.82552247517; 31.69989510732]
Centroid: CG[24.94217415839; 10.56663170244]
Coordinates of the circumscribed circle: U[21; 11.09990965907]
Coordinates of the inscribed circle: I[27.31662161584; 11.03664677186]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 106.1422275913° = 106°8'32″ = 1.28990604633 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 117.8587724087° = 117°51'28″ = 1.08545873194 rad

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How did we calculate this triangle?

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

b = 33 ; ; c = 42 ; ; beta = 44° ; ; : 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 ; ; ; ; 33**2 = 42**2 + a**2 - 2 * 42 * a * cos 44° ; ; ; ; ; ; a**2 -60.425a +675 =0 ; ; p=1; q=-60.425; r=675 ; ; D = q**2 - 4pr = 60.425**2 - 4 * 1 * 675 = 951.125424366 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.42 ± sqrt{ 951.13 } }{ 2 } ; ; : Nr. 1
a_{1,2} = 30.21227161 ± 15.4201607025 ; ; a_{1} = 45.6324323167 ; ; a_{2} = 14.7921109117 ; ; ; ; text{ Factored form: } ; ; (a -45.6324323167) (a -14.7921109117) = 0 ; ; ; ; a > 0 ; ; ; ; a = 45.632 ; ; : 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 = 45.63 ; ; b = 33 ; ; c = 42 ; ;

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

p = a+b+c = 45.63+33+42 = 120.63 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 120.63 }{ 2 } = 60.32 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.32 * (60.32-45.63)(60.32-33)(60.32-42) } ; ; T = sqrt{ 443127.16 } = 665.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 665.68 }{ 45.63 } = 29.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 665.68 }{ 33 } = 40.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 665.68 }{ 42 } = 31.7 ; ;

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{ 33**2+42**2-45.63**2 }{ 2 * 33 * 42 } ) = 73° 51'28" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 45.63**2+42**2-33**2 }{ 2 * 45.63 * 42 } ) = 44° ; ;
 gamma = 180° - alpha - beta = 180° - 73° 51'28" - 44° = 62° 8'32" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 665.68 }{ 60.32 } = 11.04 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 45.63 }{ 2 * sin 73° 51'28" } = 23.75 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 42**2 - 45.63**2 } }{ 2 } = 30.099 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 45.63**2 - 33**2 } }{ 2 } = 40.631 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 45.63**2 - 42**2 } }{ 2 } = 33.833 ; ;
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