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=11.2; b=13.4; c=2.45771059396 and a=11.2; b=13.4; c=22.02659123252.

#1 Obtuse scalene triangle.

Sides: a = 11.2   b = 13.4   c = 2.45771059396

Area: T = 6.69659466445
Perimeter: p = 27.05771059396
Semiperimeter: s = 13.52985529698

Angle ∠ A = α = 24° = 0.41988790205 rad
Angle ∠ B = β = 150.8810591188° = 150°52'50″ = 2.63333630936 rad
Angle ∠ C = γ = 5.11994088121° = 5°7'10″ = 0.08993505395 rad

Height: ha = 1.19657047579
Height: hb = 0.99993950216
Height: hc = 5.45502710172

Median: ma = 7.83882832814
Median: mb = 4.5666036005
Median: mc = 12.28878255847

Inradius: r = 0.49549492129
Circumradius: R = 13.76881226792

Vertex coordinates: A[2.45771059396; 0] B[0; 0] C[-9.78444031928; 5.45502710172]
Centroid: CG[-2.44224324177; 1.81767570057]
Coordinates of the circumscribed circle: U[1.22985529698; 13.71332001995]
Coordinates of the inscribed circle: I[0.12985529698; 0.49549492129]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156° = 0.41988790205 rad
∠ B' = β' = 29.11994088121° = 29°7'10″ = 2.63333630936 rad
∠ C' = γ' = 174.8810591188° = 174°52'50″ = 0.08993505395 rad




How did we calculate this triangle?

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

a = 11.2 ; ; b = 13.4 ; ; alpha = 24° ; ;

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 ; ; ; ; 11.2**2 = 13.4**2 + c**2 - 2 * 13.4 * c * cos(24° ) ; ; ; ; ; ; c**2 -24.483c +54.12 =0 ; ; a=1; b=-24.483; c=54.12 ; ; D = b**2 - 4ac = 24.483**2 - 4 * 1 * 54.12 = 382.938183356 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 24.48 ± sqrt{ 382.94 } }{ 2 } ; ; c_{1,2} = 12.24150913 ± 9.78440319278 ; ; c_{1} = 22.0259123228 ; ; c_{2} = 2.45710593722 ; ;
 ; ; text{ Factored form: } ; ; (c -22.0259123228) (c -2.45710593722) = 0 ; ; ; ; c > 0 ; ; ; ; c = 22.026 ; ;


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 = 11.2 ; ; b = 13.4 ; ; c = 2.46 ; ;

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

p = a+b+c = 11.2+13.4+2.46 = 27.06 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.06 }{ 2 } = 13.53 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.53 * (13.53-11.2)(13.53-13.4)(13.53-2.46) } ; ; T = sqrt{ 44.84 } = 6.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.7 }{ 11.2 } = 1.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.7 }{ 13.4 } = 1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.7 }{ 2.46 } = 5.45 ; ;

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{ 13.4**2+2.46**2-11.2**2 }{ 2 * 13.4 * 2.46 } ) = 24° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.2**2+2.46**2-13.4**2 }{ 2 * 11.2 * 2.46 } ) = 150° 52'50" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 11.2**2+13.4**2-2.46**2 }{ 2 * 11.2 * 13.4 } ) = 5° 7'10" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.7 }{ 13.53 } = 0.49 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.2 }{ 2 * sin 24° } = 13.77 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.4**2+2 * 2.46**2 - 11.2**2 } }{ 2 } = 7.838 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.46**2+2 * 11.2**2 - 13.4**2 } }{ 2 } = 4.566 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.4**2+2 * 11.2**2 - 2.46**2 } }{ 2 } = 12.288 ; ;







#2 Obtuse scalene triangle.

Sides: a = 11.2   b = 13.4   c = 22.02659123252

Area: T = 60.02435957869
Perimeter: p = 46.62659123252
Semiperimeter: s = 23.31329561626

Angle ∠ A = α = 24° = 0.41988790205 rad
Angle ∠ B = β = 29.11994088121° = 29°7'10″ = 0.508822956 rad
Angle ∠ C = γ = 126.8810591188° = 126°52'50″ = 2.21444840731 rad

Height: ha = 10.71884992477
Height: hb = 8.95987456398
Height: hc = 5.45502710172

Median: ma = 17.34990751015
Median: mb = 16.13769268102
Median: mc = 5.58770203652

Inradius: r = 2.57546883136
Circumradius: R = 13.76881226792

Vertex coordinates: A[22.02659123252; 0] B[0; 0] C[9.78444031928; 5.45502710172]
Centroid: CG[10.6033438506; 1.81767570057]
Coordinates of the circumscribed circle: U[11.01329561626; -8.26329291822]
Coordinates of the inscribed circle: I[9.91329561626; 2.57546883136]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156° = 0.41988790205 rad
∠ B' = β' = 150.8810591188° = 150°52'50″ = 0.508822956 rad
∠ C' = γ' = 53.11994088121° = 53°7'10″ = 2.21444840731 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 11.2 ; ; b = 13.4 ; ; alpha = 24° ; ; : 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 ; ; ; ; 11.2**2 = 13.4**2 + c**2 - 2 * 13.4 * c * cos(24° ) ; ; ; ; ; ; c**2 -24.483c +54.12 =0 ; ; a=1; b=-24.483; c=54.12 ; ; D = b**2 - 4ac = 24.483**2 - 4 * 1 * 54.12 = 382.938183356 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 24.48 ± sqrt{ 382.94 } }{ 2 } ; ; c_{1,2} = 12.24150913 ± 9.78440319278 ; ; c_{1} = 22.0259123228 ; ; c_{2} = 2.45710593722 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -22.0259123228) (c -2.45710593722) = 0 ; ; ; ; c > 0 ; ; ; ; c = 22.026 ; ; : 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 = 11.2 ; ; b = 13.4 ; ; c = 22.03 ; ;

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

p = a+b+c = 11.2+13.4+22.03 = 46.63 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46.63 }{ 2 } = 23.31 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.31 * (23.31-11.2)(23.31-13.4)(23.31-22.03) } ; ; T = sqrt{ 3602.83 } = 60.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 60.02 }{ 11.2 } = 10.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 60.02 }{ 13.4 } = 8.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 60.02 }{ 22.03 } = 5.45 ; ;

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{ 13.4**2+22.03**2-11.2**2 }{ 2 * 13.4 * 22.03 } ) = 24° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.2**2+22.03**2-13.4**2 }{ 2 * 11.2 * 22.03 } ) = 29° 7'10" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 11.2**2+13.4**2-22.03**2 }{ 2 * 11.2 * 13.4 } ) = 126° 52'50" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 60.02 }{ 23.31 } = 2.57 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.2 }{ 2 * sin 24° } = 13.77 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.4**2+2 * 22.03**2 - 11.2**2 } }{ 2 } = 17.349 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.03**2+2 * 11.2**2 - 13.4**2 } }{ 2 } = 16.137 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.4**2+2 * 11.2**2 - 22.03**2 } }{ 2 } = 5.587 ; ;
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