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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

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 = 12.3 ; ; b = 15.6 ; ; c = 18.6 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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




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