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=38; b=42; c=5.74663686918 and a=38; b=42; c=55.68773422442.

#1 Obtuse scalene triangle.

Sides: a = 38   b = 42   c = 5.74663686918

Area: T = 82.29992945061
Perimeter: p = 85.74663686918
Semiperimeter: s = 42.87331843459

Angle ∠ A = α = 43° = 0.75504915784 rad
Angle ∠ B = β = 131.0880449263° = 131°4'50″ = 2.28877854246 rad
Angle ∠ C = γ = 5.92195507372° = 5°55'10″ = 0.10333156506 rad

Height: ha = 4.33215418161
Height: hb = 3.91990140241
Height: hc = 28.64439311226

Median: ma = 23.18442700245
Median: mb = 17.24884891098
Median: mc = 39.94767747348

Inradius: r = 1.92195983634
Circumradius: R = 27.85993045272

Vertex coordinates: A[5.74663686918; 0] B[0; 0] C[-24.97704867762; 28.64439311226]
Centroid: CG[-6.40880393615; 9.54879770409]
Coordinates of the circumscribed circle: U[2.87331843459; 27.71107499078]
Coordinates of the inscribed circle: I[0.87331843459; 1.92195983634]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137° = 0.75504915784 rad
∠ B' = β' = 48.92195507372° = 48°55'10″ = 2.28877854246 rad
∠ C' = γ' = 174.0880449263° = 174°4'50″ = 0.10333156506 rad




How did we calculate this triangle?

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

a = 38 ; ; b = 42 ; ; alpha = 43° ; ;

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 38**2 = 42**2 + c**2 - 2 * 42 * c * cos(43° ) ; ; ; ; ; ; c**2 -61.434c +320 =0 ; ; a=1; b=-61.434; c=320 ; ; D = b**2 - 4ac = 61.434**2 - 4 * 1 * 320 = 2494.10083937 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 61.43 ± sqrt{ 2494.1 } }{ 2 } ; ; c_{1,2} = 30.71685547 ± 24.9704867762 ; ; c_{1} = 55.6873422462 ; ; c_{2} = 5.74636869376 ; ; ; ;
(c -55.6873422462) (c -5.74636869376) = 0 ; ; ; ; c > 0 ; ; ; ; c = 55.687 ; ;


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 = 38 ; ; b = 42 ; ; c = 5.75 ; ;

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

p = a+b+c = 38+42+5.75 = 85.75 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 85.75 }{ 2 } = 42.87 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.87 * (42.87-38)(42.87-42)(42.87-5.75) } ; ; T = sqrt{ 6773.17 } = 82.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82.3 }{ 38 } = 4.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82.3 }{ 42 } = 3.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82.3 }{ 5.75 } = 28.64 ; ;

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{ 38**2-42**2-5.75**2 }{ 2 * 42 * 5.75 } ) = 43° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-38**2-5.75**2 }{ 2 * 38 * 5.75 } ) = 131° 4'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.75**2-38**2-42**2 }{ 2 * 42 * 38 } ) = 5° 55'10" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82.3 }{ 42.87 } = 1.92 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38 }{ 2 * sin 43° } = 27.86 ; ;





#2 Acute scalene triangle.

Sides: a = 38   b = 42   c = 55.68773422442

Area: T = 797.5522197823
Perimeter: p = 135.6877342244
Semiperimeter: s = 67.84436711221

Angle ∠ A = α = 43° = 0.75504915784 rad
Angle ∠ B = β = 48.92195507372° = 48°55'10″ = 0.8543807229 rad
Angle ∠ C = γ = 88.08804492628° = 88°4'50″ = 1.53772938463 rad

Height: ha = 41.97664314644
Height: hb = 37.97986760868
Height: hc = 28.64439311226

Median: ma = 45.51441740902
Median: mb = 42.79664956873
Median: mc = 28.78876705977

Inradius: r = 11.75657346858
Circumradius: R = 27.85993045272

Vertex coordinates: A[55.68773422442; 0] B[0; 0] C[24.97704867762; 28.64439311226]
Centroid: CG[26.88659430068; 9.54879770409]
Coordinates of the circumscribed circle: U[27.84436711221; 0.93331812149]
Coordinates of the inscribed circle: I[25.84436711221; 11.75657346858]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137° = 0.75504915784 rad
∠ B' = β' = 131.0880449263° = 131°4'50″ = 0.8543807229 rad
∠ C' = γ' = 91.92195507372° = 91°55'10″ = 1.53772938463 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 38 ; ; b = 42 ; ; alpha = 43° ; ; : Nr. 1

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 38**2 = 42**2 + c**2 - 2 * 42 * c * cos(43° ) ; ; ; ; ; ; c**2 -61.434c +320 =0 ; ; a=1; b=-61.434; c=320 ; ; D = b**2 - 4ac = 61.434**2 - 4 * 1 * 320 = 2494.10083937 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 61.43 ± sqrt{ 2494.1 } }{ 2 } ; ; c_{1,2} = 30.71685547 ± 24.9704867762 ; ; c_{1} = 55.6873422462 ; ; c_{2} = 5.74636869376 ; ; ; ; : Nr. 1
(c -55.6873422462) (c -5.74636869376) = 0 ; ; ; ; c > 0 ; ; ; ; c = 55.687 ; ; : 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 = 38 ; ; b = 42 ; ; c = 55.69 ; ;

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

p = a+b+c = 38+42+55.69 = 135.69 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 135.69 }{ 2 } = 67.84 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 67.84 * (67.84-38)(67.84-42)(67.84-55.69) } ; ; T = sqrt{ 636089.51 } = 797.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 797.55 }{ 38 } = 41.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 797.55 }{ 42 } = 37.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 797.55 }{ 55.69 } = 28.64 ; ;

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{ 38**2-42**2-55.69**2 }{ 2 * 42 * 55.69 } ) = 43° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-38**2-55.69**2 }{ 2 * 38 * 55.69 } ) = 48° 55'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 55.69**2-38**2-42**2 }{ 2 * 42 * 38 } ) = 88° 4'50" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 797.55 }{ 67.84 } = 11.76 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 38 }{ 2 * sin 43° } = 27.86 ; ; : Nr. 1




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