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=9.70985809527; b=47; c=39 and a=70.86551453133; b=47; c=39.

#1 Obtuse scalene triangle.

Sides: a = 9.70985809527   b = 47   c = 39

Area: T = 117.5076787834
Perimeter: p = 95.70985809527
Semiperimeter: s = 47.85442904764

Angle ∠ A = α = 7.36663137074° = 7°21'59″ = 0.12985664279 rad
Angle ∠ B = β = 141.6343686293° = 141°38'1″ = 2.47219741575 rad
Angle ∠ C = γ = 31° = 0.54110520681 rad

Height: ha = 24.20767895208
Height: hb = 55.000288844
Height: hc = 6.02659891197

Median: ma = 42.91219547908
Median: mb = 15.9810559191
Median: mc = 27.77436974862

Inradius: r = 2.45655120693
Circumradius: R = 37.8611278515

Vertex coordinates: A[39; 0] B[0; 0] C[-7.61220955883; 6.02659891197]
Centroid: CG[10.46326348039; 2.00986630399]
Coordinates of the circumscribed circle: U[19.5; 32.45334499058]
Coordinates of the inscribed circle: I[0.85442904764; 2.45655120693]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172.6343686293° = 172°38'1″ = 0.12985664279 rad
∠ B' = β' = 38.36663137074° = 38°21'59″ = 2.47219741575 rad
∠ C' = γ' = 149° = 0.54110520681 rad




How did we calculate this triangle?

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

b = 47 ; ; c = 39 ; ; gamma = 31° ; ;

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 39**2 = 47**2 + a**2 - 2 * 47 * a * cos(31° ) ; ; ; ; ; ; a**2 -80.574a +688 =0 ; ; a=1; b=-80.574; c=688 ; ; D = b**2 - 4ac = 80.574**2 - 4 * 1 * 688 = 3740.12536439 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 80.57 ± sqrt{ 3740.13 } }{ 2 } ; ; a_{1,2} = 40.28686313 ± 30.5782821803 ; ; a_{1} = 70.8651453103 ; ; a_{2} = 9.70858094972 ; ; ; ;
(a -70.8651453103) (a -9.70858094972) = 0 ; ; ; ; a > 0 ; ; ; ; a = 70.865 ; ;


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 = 9.71 ; ; b = 47 ; ; c = 39 ; ;

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

p = a+b+c = 9.71+47+39 = 95.71 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 95.71 }{ 2 } = 47.85 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 47.85 * (47.85-9.71)(47.85-47)(47.85-39) } ; ; T = sqrt{ 13807.85 } = 117.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 117.51 }{ 9.71 } = 24.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 117.51 }{ 47 } = 5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 117.51 }{ 39 } = 6.03 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 9.71**2-47**2-39**2 }{ 2 * 47 * 39 } ) = 7° 21'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 47**2-9.71**2-39**2 }{ 2 * 9.71 * 39 } ) = 141° 38'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 39**2-9.71**2-47**2 }{ 2 * 47 * 9.71 } ) = 31° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 117.51 }{ 47.85 } = 2.46 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.71 }{ 2 * sin 7° 21'59" } = 37.86 ; ;





#2 Obtuse scalene triangle.

Sides: a = 70.86551453133   b = 47   c = 39

Area: T = 857.7098828479
Perimeter: p = 156.8655145313
Semiperimeter: s = 78.43325726566

Angle ∠ A = α = 110.6343686293° = 110°38'1″ = 1.93109220894 rad
Angle ∠ B = β = 38.36663137074° = 38°21'59″ = 0.6769618496 rad
Angle ∠ C = γ = 31° = 0.54110520681 rad

Height: ha = 24.20767895208
Height: hb = 36.49882480204
Height: hc = 43.98550681271

Median: ma = 24.68987179686
Median: mb = 52.14657995445
Median: mc = 56.87986815084

Inradius: r = 10.93656202331
Circumradius: R = 37.8611278515

Vertex coordinates: A[39; 0] B[0; 0] C[55.56224207727; 43.98550681271]
Centroid: CG[31.52108069242; 14.66216893757]
Coordinates of the circumscribed circle: U[19.5; 32.45334499058]
Coordinates of the inscribed circle: I[31.43325726566; 10.93656202331]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 69.36663137074° = 69°21'59″ = 1.93109220894 rad
∠ B' = β' = 141.6343686293° = 141°38'1″ = 0.6769618496 rad
∠ C' = γ' = 149° = 0.54110520681 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 47 ; ; c = 39 ; ; gamma = 31° ; ; : Nr. 1

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 39**2 = 47**2 + a**2 - 2 * 47 * a * cos(31° ) ; ; ; ; ; ; a**2 -80.574a +688 =0 ; ; a=1; b=-80.574; c=688 ; ; D = b**2 - 4ac = 80.574**2 - 4 * 1 * 688 = 3740.12536439 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 80.57 ± sqrt{ 3740.13 } }{ 2 } ; ; a_{1,2} = 40.28686313 ± 30.5782821803 ; ; a_{1} = 70.8651453103 ; ; a_{2} = 9.70858094972 ; ; ; ; : Nr. 1
(a -70.8651453103) (a -9.70858094972) = 0 ; ; ; ; a > 0 ; ; ; ; a = 70.865 ; ; : 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 = 70.87 ; ; b = 47 ; ; c = 39 ; ;

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

p = a+b+c = 70.87+47+39 = 156.87 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 156.87 }{ 2 } = 78.43 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.43 * (78.43-70.87)(78.43-47)(78.43-39) } ; ; T = sqrt{ 735664.43 } = 857.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 857.71 }{ 70.87 } = 24.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 857.71 }{ 47 } = 36.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 857.71 }{ 39 } = 43.99 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 70.87**2-47**2-39**2 }{ 2 * 47 * 39 } ) = 110° 38'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 47**2-70.87**2-39**2 }{ 2 * 70.87 * 39 } ) = 38° 21'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 39**2-70.87**2-47**2 }{ 2 * 47 * 70.87 } ) = 31° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 857.71 }{ 78.43 } = 10.94 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 70.87 }{ 2 * sin 110° 38'1" } = 37.86 ; ;




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