Triangle calculator

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

You have entered side a, b and angle α.

Triangle has two solutions: a=12.3; b=15.6; c=4.95111104075 and a=12.3; b=15.6; c=18.59658284954.

#1 Obtuse scalene triangle.

Sides: a = 12.3   b = 15.6   c = 4.95111104075

Area: T = 25.33661213537
Perimeter: p = 32.85111104075
Semiperimeter: s = 16.42655552038

Angle ∠ A = α = 41° = 0.71655849933 rad
Angle ∠ B = β = 123.6887529396° = 123°41'15″ = 2.15987546316 rad
Angle ∠ C = γ = 15.31224706044° = 15°18'45″ = 0.26772530287 rad

Height: ha = 4.12196945291
Height: hb = 3.24882206864
Height: hc = 10.23545208523

Median: ma = 9.80437873872
Median: mb = 5.20220906503
Median: mc = 13.82773868259

Inradius: r = 1.54224818851
Circumradius: R = 9.37441564832

Vertex coordinates: A[4.95111104075; 0] B[0; 0] C[-6.8222359044; 10.23545208523]
Centroid: CG[-0.62437495455; 3.41215069508]
Coordinates of the circumscribed circle: U[2.47655552038; 9.04113735796]
Coordinates of the inscribed circle: I[0.82655552038; 1.54224818851]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139° = 0.71655849933 rad
∠ B' = β' = 56.31224706044° = 56°18'45″ = 2.15987546316 rad
∠ C' = γ' = 164.6887529396° = 164°41'15″ = 0.26772530287 rad


How did we calculate this triangle?

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

a = 12.3 ; ; b = 15.6 ; ; alpha = 41° ; ;

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 12.3**2 = 15.6**2 + c**2 - 2 * 15.6 * c * cos 41° ; ; ; ; ; ; c**2 -23.547c +92.07 =0 ; ; p=1; q=-23.547; r=92.07 ; ; D = q**2 - 4pr = 23.547**2 - 4 * 1 * 92.07 = 186.178331699 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 23.55 ± sqrt{ 186.18 } }{ 2 } ; ;
c_{1,2} = 11.77346945 ± 6.82235904397 ; ; c_{1} = 18.5958284954 ; ; c_{2} = 4.95111040751 ; ; ; ; text{ Factored form: } ; ; (c -18.5958284954) (c -4.95111040751) = 0 ; ; ; ; c > 0 ; ; ; ; c = 18.596 ; ;
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 = 12.3 ; ; b = 15.6 ; ; c = 4.95 ; ;

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

p = a+b+c = 12.3+15.6+4.95 = 32.85 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.85 }{ 2 } = 16.43 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.43 * (16.43-12.3)(16.43-15.6)(16.43-4.95) } ; ; T = sqrt{ 641.92 } = 25.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.34 }{ 12.3 } = 4.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.34 }{ 15.6 } = 3.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.34 }{ 4.95 } = 10.23 ; ;

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{ 15.6**2+4.95**2-12.3**2 }{ 2 * 15.6 * 4.95 } ) = 41° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.3**2+4.95**2-15.6**2 }{ 2 * 12.3 * 4.95 } ) = 123° 41'15" ; ;
 gamma = 180° - alpha - beta = 180° - 41° - 123° 41'15" = 15° 18'45" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.34 }{ 16.43 } = 1.54 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.3 }{ 2 * sin 41° } = 9.37 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.6**2+2 * 4.95**2 - 12.3**2 } }{ 2 } = 9.804 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.95**2+2 * 12.3**2 - 15.6**2 } }{ 2 } = 5.202 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.6**2+2 * 12.3**2 - 4.95**2 } }{ 2 } = 13.827 ; ;



#2 Acute scalene triangle.

Sides: a = 12.3   b = 15.6   c = 18.59658284954

Area: T = 95.16596972508
Perimeter: p = 46.49658284954
Semiperimeter: s = 23.24879142477

Angle ∠ A = α = 41° = 0.71655849933 rad
Angle ∠ B = β = 56.31224706044° = 56°18'45″ = 0.9832838022 rad
Angle ∠ C = γ = 82.68875293956° = 82°41'15″ = 1.44331696383 rad

Height: ha = 15.47331215042
Height: hb = 12.2199961186
Height: hc = 10.23545208523

Median: ma = 16.02437298628
Median: mb = 13.70106357048
Median: mc = 10.53296624182

Inradius: r = 4.09332574095
Circumradius: R = 9.37441564832

Vertex coordinates: A[18.59658284954; 0] B[0; 0] C[6.8222359044; 10.23545208523]
Centroid: CG[8.47327291798; 3.41215069508]
Coordinates of the circumscribed circle: U[9.29879142477; 1.19331472727]
Coordinates of the inscribed circle: I[7.64879142477; 4.09332574095]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139° = 0.71655849933 rad
∠ B' = β' = 123.6887529396° = 123°41'15″ = 0.9832838022 rad
∠ C' = γ' = 97.31224706044° = 97°18'45″ = 1.44331696383 rad

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How did we calculate this triangle?

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

a = 12.3 ; ; b = 15.6 ; ; alpha = 41° ; ; : Nr. 1

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 12.3**2 = 15.6**2 + c**2 - 2 * 15.6 * c * cos 41° ; ; ; ; ; ; c**2 -23.547c +92.07 =0 ; ; p=1; q=-23.547; r=92.07 ; ; D = q**2 - 4pr = 23.547**2 - 4 * 1 * 92.07 = 186.178331699 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 23.55 ± sqrt{ 186.18 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 11.77346945 ± 6.82235904397 ; ; c_{1} = 18.5958284954 ; ; c_{2} = 4.95111040751 ; ; ; ; text{ Factored form: } ; ; (c -18.5958284954) (c -4.95111040751) = 0 ; ; ; ; c > 0 ; ; ; ; c = 18.596 ; ; : 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 = 12.3 ; ; b = 15.6 ; ; c = 18.6 ; ;

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

p = a+b+c = 12.3+15.6+18.6 = 46.5 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46.5 }{ 2 } = 23.25 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.25 * (23.25-12.3)(23.25-15.6)(23.25-18.6) } ; ; T = sqrt{ 9055.37 } = 95.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 95.16 }{ 12.3 } = 15.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 95.16 }{ 15.6 } = 12.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 95.16 }{ 18.6 } = 10.23 ; ;

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{ 15.6**2+18.6**2-12.3**2 }{ 2 * 15.6 * 18.6 } ) = 41° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.3**2+18.6**2-15.6**2 }{ 2 * 12.3 * 18.6 } ) = 56° 18'45" ; ;
 gamma = 180° - alpha - beta = 180° - 41° - 56° 18'45" = 82° 41'15" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 95.16 }{ 23.25 } = 4.09 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.3 }{ 2 * sin 41° } = 9.37 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.6**2+2 * 18.6**2 - 12.3**2 } }{ 2 } = 16.024 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.6**2+2 * 12.3**2 - 15.6**2 } }{ 2 } = 13.701 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.6**2+2 * 12.3**2 - 18.6**2 } }{ 2 } = 10.53 ; ;
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