Triangle calculator

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

You have entered side b, c and angle γ.

Triangle has two solutions: a=2.49329557658; b=20.2; c=18.3 and a=29.34326786799; b=20.2; c=18.3.

#1 Obtuse scalene triangle.

Sides: a = 2.49329557658   b = 20.2   c = 18.3

Area: T = 15.50216499296
Perimeter: p = 40.99329557658
Semiperimeter: s = 20.49664778829

Angle ∠ A = α = 4.81110354856° = 4°48'40″ = 0.08439684097 rad
Angle ∠ B = β = 137.1898964514° = 137°11'20″ = 2.39443991282 rad
Angle ∠ C = γ = 38° = 0.66332251158 rad

Height: ha = 12.43663618016
Height: hb = 1.53548168247
Height: hc = 1.69441693912

Median: ma = 19.23330780919
Median: mb = 8.27990346192
Median: mc = 11.10987764504

Inradius: r = 0.75663079871
Circumradius: R = 14.86220635962

Vertex coordinates: A[18.3; 0] B[0; 0] C[-1.82988298238; 1.69441693912]
Centroid: CG[5.49903900587; 0.56547231304]
Coordinates of the circumscribed circle: U[9.15; 11.71114659346]
Coordinates of the inscribed circle: I[0.29664778829; 0.75663079871]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 175.1898964514° = 175°11'20″ = 0.08439684097 rad
∠ B' = β' = 42.81110354856° = 42°48'40″ = 2.39443991282 rad
∠ C' = γ' = 142° = 0.66332251158 rad




How did we calculate this triangle?

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

b = 20.2 ; ; c = 18.3 ; ; gamma = 38° ; ;

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 18.3**2 = 20.2**2 + a**2 - 2 * 20.2 * a * cos 38° ; ; ; ; ; ; a**2 -31.836a +73.15 =0 ; ; a=1; b=-31.836; c=73.15 ; ; D = b**2 - 4ac = 31.836**2 - 4 * 1 * 73.15 = 720.907620561 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 31.84 ± sqrt{ 720.91 } }{ 2 } ; ; a_{1,2} = 15.91781722 ± 13.424861457 ; ; a_{1} = 29.342678677 ; ; a_{2} = 2.49295576298 ; ;
 ; ; text{ Factored form: } ; ; (a -29.342678677) (a -2.49295576298) = 0 ; ; ; ; a > 0 ; ; ; ; a = 29.343 ; ;


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 = 2.49 ; ; b = 20.2 ; ; c = 18.3 ; ;

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

p = a+b+c = 2.49+20.2+18.3 = 40.99 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.99 }{ 2 } = 20.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.5 * (20.5-2.49)(20.5-20.2)(20.5-18.3) } ; ; T = sqrt{ 240.3 } = 15.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15.5 }{ 2.49 } = 12.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15.5 }{ 20.2 } = 1.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15.5 }{ 18.3 } = 1.69 ; ;

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{ 20.2**2+18.3**2-2.49**2 }{ 2 * 20.2 * 18.3 } ) = 4° 48'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.49**2+18.3**2-20.2**2 }{ 2 * 2.49 * 18.3 } ) = 137° 11'20" ; ; gamma = 180° - alpha - beta = 180° - 4° 48'40" - 137° 11'20" = 38° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15.5 }{ 20.5 } = 0.76 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.49 }{ 2 * sin 4° 48'40" } = 14.86 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.2**2+2 * 18.3**2 - 2.49**2 } }{ 2 } = 19.233 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.3**2+2 * 2.49**2 - 20.2**2 } }{ 2 } = 8.279 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.2**2+2 * 2.49**2 - 18.3**2 } }{ 2 } = 11.109 ; ;







#2 Obtuse scalene triangle.

Sides: a = 29.34326786799   b = 20.2   c = 18.3

Area: T = 182.4588084145
Perimeter: p = 67.84326786799
Semiperimeter: s = 33.92113393399

Angle ∠ A = α = 99.18989645144° = 99°11'20″ = 1.73111740124 rad
Angle ∠ B = β = 42.81110354856° = 42°48'40″ = 0.74771935254 rad
Angle ∠ C = γ = 38° = 0.66332251158 rad

Height: ha = 12.43663618016
Height: hb = 18.06551568461
Height: hc = 19.94107742235

Median: ma = 12.49986720084
Median: mb = 22.2769517194
Median: mc = 23.46989986164

Inradius: r = 5.37988584913
Circumradius: R = 14.86220635962

Vertex coordinates: A[18.3; 0] B[0; 0] C[21.52657593473; 19.94107742235]
Centroid: CG[13.27552531158; 6.64769247412]
Coordinates of the circumscribed circle: U[9.15; 11.71114659346]
Coordinates of the inscribed circle: I[13.72113393399; 5.37988584913]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 80.81110354856° = 80°48'40″ = 1.73111740124 rad
∠ B' = β' = 137.1898964514° = 137°11'20″ = 0.74771935254 rad
∠ C' = γ' = 142° = 0.66332251158 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 20.2 ; ; c = 18.3 ; ; gamma = 38° ; ; : Nr. 1

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 18.3**2 = 20.2**2 + a**2 - 2 * 20.2 * a * cos 38° ; ; ; ; ; ; a**2 -31.836a +73.15 =0 ; ; a=1; b=-31.836; c=73.15 ; ; D = b**2 - 4ac = 31.836**2 - 4 * 1 * 73.15 = 720.907620561 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 31.84 ± sqrt{ 720.91 } }{ 2 } ; ; a_{1,2} = 15.91781722 ± 13.424861457 ; ; a_{1} = 29.342678677 ; ; a_{2} = 2.49295576298 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (a -29.342678677) (a -2.49295576298) = 0 ; ; ; ; a > 0 ; ; ; ; a = 29.343 ; ; : 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 = 29.34 ; ; b = 20.2 ; ; c = 18.3 ; ;

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

p = a+b+c = 29.34+20.2+18.3 = 67.84 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67.84 }{ 2 } = 33.92 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.92 * (33.92-29.34)(33.92-20.2)(33.92-18.3) } ; ; T = sqrt{ 33290.95 } = 182.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 182.46 }{ 29.34 } = 12.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 182.46 }{ 20.2 } = 18.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 182.46 }{ 18.3 } = 19.94 ; ;

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{ 20.2**2+18.3**2-29.34**2 }{ 2 * 20.2 * 18.3 } ) = 99° 11'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 29.34**2+18.3**2-20.2**2 }{ 2 * 29.34 * 18.3 } ) = 42° 48'40" ; ; gamma = 180° - alpha - beta = 180° - 99° 11'20" - 42° 48'40" = 38° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 182.46 }{ 33.92 } = 5.38 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 29.34 }{ 2 * sin 99° 11'20" } = 14.86 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.2**2+2 * 18.3**2 - 29.34**2 } }{ 2 } = 12.499 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.3**2+2 * 29.34**2 - 20.2**2 } }{ 2 } = 22.27 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.2**2+2 * 29.34**2 - 18.3**2 } }{ 2 } = 23.469 ; ;
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