Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=38; b=25.57875251515; c=49 and a=38; b=37.41656605978; c=49.

#1 Obtuse scalene triangle.

Sides: a = 38   b = 25.57875251515   c = 49

Area: T = 480.0411264771
Perimeter: p = 112.5787525151
Semiperimeter: s = 56.28987625758

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 31.03988370467° = 31°2'20″ = 0.54217299025 rad
Angle ∠ C = γ = 98.96111629533° = 98°57'40″ = 1.72771981251 rad

Height: ha = 25.26553297248
Height: hb = 37.53661777128
Height: hc = 19.59435210111

Median: ma = 34.15655983177
Median: mb = 41.9439808676
Median: mc = 21.18661958935

Inradius: r = 8.52881900473
Circumradius: R = 24.80327384973

Vertex coordinates: A[49; 0] B[0; 0] C[32.55990837462; 19.59435210111]
Centroid: CG[27.18663612487; 6.53111736704]
Coordinates of the circumscribed circle: U[24.5; -3.86333970759]
Coordinates of the inscribed circle: I[30.71112374242; 8.52881900473]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 148.9611162953° = 148°57'40″ = 0.54217299025 rad
∠ C' = γ' = 81.03988370467° = 81°2'20″ = 1.72771981251 rad




How did we calculate this triangle?

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

a = 38 ; ; c = 49 ; ; alpha = 50° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 38**2 = 49**2 + b**2 - 2 * 49 * b * cos 50° ; ; ; ; ; ; b**2 -62.993b +957 =0 ; ; p=1; q=-62.993; r=957 ; ; D = q**2 - 4pr = 62.993**2 - 4 * 1 * 957 = 140.141450843 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.99 ± sqrt{ 140.14 } }{ 2 } ; ; b_{1,2} = 31.49659287 ± 5.91906772312 ; ; b_{1} = 37.4156605931 ; ; b_{2} = 25.5775251469 ; ; ; ; text{ Factored form: } ; ; (b -37.4156605931) (b -25.5775251469) = 0 ; ; ; ; b > 0 ; ; ; ; b = 37.416 ; ;


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 = 25.58 ; ; c = 49 ; ;

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

p = a+b+c = 38+25.58+49 = 112.58 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 112.58 }{ 2 } = 56.29 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 56.29 * (56.29-38)(56.29-25.58)(56.29-49) } ; ; T = sqrt{ 230439.62 } = 480.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 480.04 }{ 38 } = 25.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 480.04 }{ 25.58 } = 37.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 480.04 }{ 49 } = 19.59 ; ;

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{ 25.58**2+49**2-38**2 }{ 2 * 25.58 * 49 } ) = 50° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38**2+49**2-25.58**2 }{ 2 * 38 * 49 } ) = 31° 2'20" ; ; gamma = 180° - alpha - beta = 180° - 50° - 31° 2'20" = 98° 57'40" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 480.04 }{ 56.29 } = 8.53 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 38 }{ 2 * sin 50° } = 24.8 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.58**2+2 * 49**2 - 38**2 } }{ 2 } = 34.156 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 38**2 - 25.58**2 } }{ 2 } = 41.94 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.58**2+2 * 38**2 - 49**2 } }{ 2 } = 21.186 ; ;







#2 Acute scalene triangle.

Sides: a = 38   b = 37.41656605978   c = 49

Area: T = 702.222044272
Perimeter: p = 124.4165660598
Semiperimeter: s = 62.20878302989

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 48.96111629533° = 48°57'40″ = 0.85545334991 rad
Angle ∠ C = γ = 81.03988370467° = 81°2'20″ = 1.41443945285 rad

Height: ha = 36.95989706695
Height: hb = 37.53661777128
Height: hc = 28.66220588865

Median: ma = 39.23660271814
Median: mb = 39.65549755454
Median: mc = 28.66655861441

Inradius: r = 11.28882966557
Circumradius: R = 24.80327384973

Vertex coordinates: A[49; 0] B[0; 0] C[24.95496769595; 28.66220588865]
Centroid: CG[24.65498923198; 9.55440196288]
Coordinates of the circumscribed circle: U[24.5; 3.86333970759]
Coordinates of the inscribed circle: I[24.79221697011; 11.28882966557]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 131.0398837047° = 131°2'20″ = 0.85545334991 rad
∠ C' = γ' = 98.96111629533° = 98°57'40″ = 1.41443945285 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 38 ; ; c = 49 ; ; alpha = 50° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 38**2 = 49**2 + b**2 - 2 * 49 * b * cos 50° ; ; ; ; ; ; b**2 -62.993b +957 =0 ; ; p=1; q=-62.993; r=957 ; ; D = q**2 - 4pr = 62.993**2 - 4 * 1 * 957 = 140.141450843 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.99 ± sqrt{ 140.14 } }{ 2 } ; ; b_{1,2} = 31.49659287 ± 5.91906772312 ; ; b_{1} = 37.4156605931 ; ; b_{2} = 25.5775251469 ; ; ; ; text{ Factored form: } ; ; (b -37.4156605931) (b -25.5775251469) = 0 ; ; ; ; b > 0 ; ; ; ; b = 37.416 ; ; : 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 = 37.42 ; ; c = 49 ; ;

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

p = a+b+c = 38+37.42+49 = 124.42 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 124.42 }{ 2 } = 62.21 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 62.21 * (62.21-38)(62.21-37.42)(62.21-49) } ; ; T = sqrt{ 493113.55 } = 702.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 702.22 }{ 38 } = 36.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 702.22 }{ 37.42 } = 37.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 702.22 }{ 49 } = 28.66 ; ;

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{ 37.42**2+49**2-38**2 }{ 2 * 37.42 * 49 } ) = 50° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38**2+49**2-37.42**2 }{ 2 * 38 * 49 } ) = 48° 57'40" ; ; gamma = 180° - alpha - beta = 180° - 50° - 48° 57'40" = 81° 2'20" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 702.22 }{ 62.21 } = 11.29 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 38 }{ 2 * sin 50° } = 24.8 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.42**2+2 * 49**2 - 38**2 } }{ 2 } = 39.236 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 49**2+2 * 38**2 - 37.42**2 } }{ 2 } = 39.655 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.42**2+2 * 38**2 - 49**2 } }{ 2 } = 28.666 ; ;
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