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=3.4; b=3.19441125497; c=4.8 and a=3.4; b=3.59441125497; c=4.8.

#1 Obtuse scalene triangle.

Sides: a = 3.4   b = 3.19441125497   c = 4.8

Area: T = 5.4210588745
Perimeter: p = 11.39441125497
Semiperimeter: s = 5.69770562748

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 41.62877133166° = 41°37'40″ = 0.72765406575 rad
Angle ∠ C = γ = 93.37222866834° = 93°22'20″ = 1.63296538327 rad

Height: ha = 3.18985816147
Height: hb = 3.39441125497
Height: hc = 2.25985786438

Median: ma = 3.70655603476
Median: mb = 3.84404962251
Median: mc = 2.26330018758

Inradius: r = 0.95114718626
Circumradius: R = 2.4044163056

Vertex coordinates: A[4.8; 0] B[0; 0] C[2.54114213562; 2.25985786438]
Centroid: CG[2.44771404521; 0.75328595479]
Coordinates of the circumscribed circle: U[2.4; -0.14114213562]
Coordinates of the inscribed circle: I[2.50329437252; 0.95114718626]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 138.3722286683° = 138°22'20″ = 0.72765406575 rad
∠ C' = γ' = 86.62877133166° = 86°37'40″ = 1.63296538327 rad




How did we calculate this triangle?

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

a = 3.4 ; ; c = 4.8 ; ; alpha = 45° ; ;

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 ; ; ; ; 3.4**2 = 4.8**2 + b**2 - 2 * 4.8 * b * cos 45° ; ; ; ; ; ; b**2 -6.788b +11.48 =0 ; ; a=1; b=-6.788; c=11.48 ; ; D = b**2 - 4ac = 6.788**2 - 4 * 1 * 11.48 = 0.16 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 6.79 ± sqrt{ 0.16 } }{ 2 } ; ; b_{1,2} = 3.39411255 ± 0.2 ; ; b_{1} = 3.59411255 ; ; b_{2} = 3.19411255 ; ; ; ; text{ Factored form: } ; ;
(b -3.59411255) (b -3.19411255) = 0 ; ; ; ; b > 0 ; ; ; ; b = 3.594 ; ;


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 = 3.4 ; ; b = 3.19 ; ; c = 4.8 ; ;

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

p = a+b+c = 3.4+3.19+4.8 = 11.39 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.39 }{ 2 } = 5.7 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.7 * (5.7-3.4)(5.7-3.19)(5.7-4.8) } ; ; T = sqrt{ 29.38 } = 5.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.42 }{ 3.4 } = 3.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.42 }{ 3.19 } = 3.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.42 }{ 4.8 } = 2.26 ; ;

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{ 3.19**2+4.8**2-3.4**2 }{ 2 * 3.19 * 4.8 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.4**2+4.8**2-3.19**2 }{ 2 * 3.4 * 4.8 } ) = 41° 37'40" ; ; gamma = 180° - alpha - beta = 180° - 45° - 41° 37'40" = 93° 22'20" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.42 }{ 5.7 } = 0.95 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.4 }{ 2 * sin 45° } = 2.4 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.19**2+2 * 4.8**2 - 3.4**2 } }{ 2 } = 3.706 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.8**2+2 * 3.4**2 - 3.19**2 } }{ 2 } = 3.84 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.19**2+2 * 3.4**2 - 4.8**2 } }{ 2 } = 2.263 ; ;







#2 Acute scalene triangle.

Sides: a = 3.4   b = 3.59441125497   c = 4.8

Area: T = 6.0999411255
Perimeter: p = 11.79441125497
Semiperimeter: s = 5.89770562748

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 48.37222866834° = 48°22'20″ = 0.84442556693 rad
Angle ∠ C = γ = 86.62877133166° = 86°37'40″ = 1.51219388208 rad

Height: ha = 3.58878889735
Height: hb = 3.39441125497
Height: hc = 2.54114213562

Median: ma = 3.88444333576
Median: mb = 3.75110783443
Median: mc = 2.54553531209

Inradius: r = 1.03443145751
Circumradius: R = 2.4044163056

Vertex coordinates: A[4.8; 0] B[0; 0] C[2.25985786438; 2.54114213562]
Centroid: CG[2.35328595479; 0.84771404521]
Coordinates of the circumscribed circle: U[2.4; 0.14114213562]
Coordinates of the inscribed circle: I[2.30329437252; 1.03443145751]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 131.6287713317° = 131°37'40″ = 0.84442556693 rad
∠ C' = γ' = 93.37222866834° = 93°22'20″ = 1.51219388208 rad

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How did we calculate this triangle?

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

a = 3.4 ; ; c = 4.8 ; ; alpha = 45° ; ; : 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 ; ; ; ; 3.4**2 = 4.8**2 + b**2 - 2 * 4.8 * b * cos 45° ; ; ; ; ; ; b**2 -6.788b +11.48 =0 ; ; a=1; b=-6.788; c=11.48 ; ; D = b**2 - 4ac = 6.788**2 - 4 * 1 * 11.48 = 0.16 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 6.79 ± sqrt{ 0.16 } }{ 2 } ; ; b_{1,2} = 3.39411255 ± 0.2 ; ; b_{1} = 3.59411255 ; ; b_{2} = 3.19411255 ; ; ; ; text{ Factored form: } ; ; : Nr. 1
(b -3.59411255) (b -3.19411255) = 0 ; ; ; ; b > 0 ; ; ; ; b = 3.594 ; ; : 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 = 3.4 ; ; b = 3.59 ; ; c = 4.8 ; ;

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

p = a+b+c = 3.4+3.59+4.8 = 11.79 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.79 }{ 2 } = 5.9 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.9 * (5.9-3.4)(5.9-3.59)(5.9-4.8) } ; ; T = sqrt{ 37.2 } = 6.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.1 }{ 3.4 } = 3.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.1 }{ 3.59 } = 3.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.1 }{ 4.8 } = 2.54 ; ;

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{ 3.59**2+4.8**2-3.4**2 }{ 2 * 3.59 * 4.8 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.4**2+4.8**2-3.59**2 }{ 2 * 3.4 * 4.8 } ) = 48° 22'20" ; ; gamma = 180° - alpha - beta = 180° - 45° - 48° 22'20" = 86° 37'40" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.1 }{ 5.9 } = 1.03 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.4 }{ 2 * sin 45° } = 2.4 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.59**2+2 * 4.8**2 - 3.4**2 } }{ 2 } = 3.884 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.8**2+2 * 3.4**2 - 3.59**2 } }{ 2 } = 3.751 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.59**2+2 * 3.4**2 - 4.8**2 } }{ 2 } = 2.545 ; ;
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