Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=16.85; b=7.35; c=10.07993227815 and a=16.85; b=7.35; c=22.80990720959.

#1 Obtuse scalene triangle.

Sides: a = 16.85   b = 7.35   c = 10.07993227815

Area: T = 18.52443520017
Perimeter: p = 34.27993227815
Semiperimeter: s = 17.14396613907

Angle ∠ A = α = 149.9943576315° = 149°59'37″ = 2.61878817635 rad
Angle ∠ B = β = 12.6° = 12°36' = 0.22199114858 rad
Angle ∠ C = γ = 17.40664236849° = 17°24'23″ = 0.30437994043 rad

Height: ha = 2.19987361426
Height: hb = 5.04106400005
Height: hc = 3.67657136175

Median: ma = 2.61328526301
Median: mb = 13.38985024878
Median: mc = 11.98222499167

Inradius: r = 1.08107886795
Circumradius: R = 16.84767286746

Vertex coordinates: A[10.07993227815; 0] B[0; 0] C[16.44441974387; 3.67657136175]
Centroid: CG[8.84111734067; 1.22552378725]
Coordinates of the circumscribed circle: U[5.04396613907; 16.0755262987]
Coordinates of the inscribed circle: I[9.79896613907; 1.08107886795]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.00664236849° = 30°23″ = 2.61878817635 rad
∠ B' = β' = 167.4° = 167°24' = 0.22199114858 rad
∠ C' = γ' = 162.5943576315° = 162°35'37″ = 0.30437994043 rad




How did we calculate this triangle?

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

a = 16.85 ; ; b = 7.35 ; ; beta = 12.6° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 7.35**2 = 16.85**2 + c**2 - 2 * 16.85 * c * cos 12° 36' ; ; ; ; ; ; c**2 -32.888c +229.9 =0 ; ; p=1; q=-32.888; r=229.9 ; ; D = q**2 - 4pr = 32.888**2 - 4 * 1 * 229.9 = 162.046517608 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 32.89 ± sqrt{ 162.05 } }{ 2 } ; ; c_{1,2} = 16.44419744 ± 6.3648746572 ; ; c_{1} = 22.8090720972 ; ; c_{2} = 10.0793227828 ; ; ; ; text{ Factored form: } ; ; (c -22.8090720972) (c -10.0793227828) = 0 ; ; ; ; c > 0 ; ; ; ; c = 22.809 ; ;


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 = 16.85 ; ; b = 7.35 ; ; c = 10.08 ; ;

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

p = a+b+c = 16.85+7.35+10.08 = 34.28 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34.28 }{ 2 } = 17.14 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.14 * (17.14-16.85)(17.14-7.35)(17.14-10.08) } ; ; T = sqrt{ 343.15 } = 18.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.52 }{ 16.85 } = 2.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.52 }{ 7.35 } = 5.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.52 }{ 10.08 } = 3.68 ; ;

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{ 7.35**2+10.08**2-16.85**2 }{ 2 * 7.35 * 10.08 } ) = 149° 59'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16.85**2+10.08**2-7.35**2 }{ 2 * 16.85 * 10.08 } ) = 12° 36' ; ; gamma = 180° - alpha - beta = 180° - 149° 59'37" - 12° 36' = 17° 24'23" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.52 }{ 17.14 } = 1.08 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16.85 }{ 2 * sin 149° 59'37" } = 16.85 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.35**2+2 * 10.08**2 - 16.85**2 } }{ 2 } = 2.613 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.08**2+2 * 16.85**2 - 7.35**2 } }{ 2 } = 13.389 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.35**2+2 * 16.85**2 - 10.08**2 } }{ 2 } = 11.982 ; ;







#2 Obtuse scalene triangle.

Sides: a = 16.85   b = 7.35   c = 22.80990720959

Area: T = 41.9219808453
Perimeter: p = 47.00990720959
Semiperimeter: s = 23.50545360479

Angle ∠ A = α = 30.00664236849° = 30°23″ = 0.524371089 rad
Angle ∠ B = β = 12.6° = 12°36' = 0.22199114858 rad
Angle ∠ C = γ = 137.3943576315° = 137°23'37″ = 2.39879702778 rad

Height: ha = 4.97656449202
Height: hb = 11.40767505995
Height: hc = 3.67657136175

Median: ma = 14.70222960771
Median: mb = 19.71224962888
Median: mc = 6.23877125239

Inradius: r = 1.78334773836
Circumradius: R = 16.84767286746

Vertex coordinates: A[22.80990720959; 0] B[0; 0] C[16.44441974387; 3.67657136175]
Centroid: CG[13.08444231782; 1.22552378725]
Coordinates of the circumscribed circle: U[11.40545360479; -12.43995493695]
Coordinates of the inscribed circle: I[16.15545360479; 1.78334773836]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.9943576315° = 149°59'37″ = 0.524371089 rad
∠ B' = β' = 167.4° = 167°24' = 0.22199114858 rad
∠ C' = γ' = 42.60664236849° = 42°36'23″ = 2.39879702778 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 16.85 ; ; b = 7.35 ; ; beta = 12.6° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 7.35**2 = 16.85**2 + c**2 - 2 * 16.85 * c * cos 12° 36' ; ; ; ; ; ; c**2 -32.888c +229.9 =0 ; ; p=1; q=-32.888; r=229.9 ; ; D = q**2 - 4pr = 32.888**2 - 4 * 1 * 229.9 = 162.046517608 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 32.89 ± sqrt{ 162.05 } }{ 2 } ; ; c_{1,2} = 16.44419744 ± 6.3648746572 ; ; c_{1} = 22.8090720972 ; ; c_{2} = 10.0793227828 ; ; ; ; text{ Factored form: } ; ; (c -22.8090720972) (c -10.0793227828) = 0 ; ; ; ; c > 0 ; ; ; ; c = 22.809 ; ; : 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 = 16.85 ; ; b = 7.35 ; ; c = 22.81 ; ;

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

p = a+b+c = 16.85+7.35+22.81 = 47.01 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47.01 }{ 2 } = 23.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-16.85)(23.5-7.35)(23.5-22.81) } ; ; T = sqrt{ 1757.27 } = 41.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.92 }{ 16.85 } = 4.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.92 }{ 7.35 } = 11.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.92 }{ 22.81 } = 3.68 ; ;

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{ 7.35**2+22.81**2-16.85**2 }{ 2 * 7.35 * 22.81 } ) = 30° 23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16.85**2+22.81**2-7.35**2 }{ 2 * 16.85 * 22.81 } ) = 12° 36' ; ; gamma = 180° - alpha - beta = 180° - 30° 23" - 12° 36' = 137° 23'37" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.92 }{ 23.5 } = 1.78 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16.85 }{ 2 * sin 30° 23" } = 16.85 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.35**2+2 * 22.81**2 - 16.85**2 } }{ 2 } = 14.702 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.81**2+2 * 16.85**2 - 7.35**2 } }{ 2 } = 19.712 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.35**2+2 * 16.85**2 - 22.81**2 } }{ 2 } = 6.238 ; ;
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