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=13.5; b=16.5; c=3.88988340668 and a=13.5; b=16.5; c=23.14331833947.

#1 Obtuse scalene triangle.

Sides: a = 13.5   b = 16.5   c = 3.88988340668

Area: T = 18.40219845812
Perimeter: p = 33.88988340668
Semiperimeter: s = 16.94444170334

Angle ∠ A = α = 35° = 0.61108652382 rad
Angle ∠ B = β = 135.4989668391° = 135°29'23″ = 2.36547408159 rad
Angle ∠ C = γ = 9.51103316092° = 9°30'37″ = 0.16659865995 rad

Height: ha = 2.7266219938
Height: hb = 2.23105435856
Height: hc = 9.46440111998

Median: ma = 9.90657566697
Median: mb = 5.53438969271
Median: mc = 14.94988876643

Inradius: r = 1.0866020519
Circumradius: R = 11.76882658704

Vertex coordinates: A[3.88988340668; 0] B[0; 0] C[-9.6277174664; 9.46440111998]
Centroid: CG[-1.9132780199; 3.15546703999]
Coordinates of the circumscribed circle: U[1.94444170334; 11.60765207533]
Coordinates of the inscribed circle: I[0.44444170334; 1.0866020519]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145° = 0.61108652382 rad
∠ B' = β' = 44.51103316092° = 44°30'37″ = 2.36547408159 rad
∠ C' = γ' = 170.4989668391° = 170°29'23″ = 0.16659865995 rad




How did we calculate this triangle?

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

a = 13.5 ; ; b = 16.5 ; ; alpha = 35° ; ;

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 ; ; ; ; 13.5**2 = 16.5**2 + c**2 - 2 * 16.5 * c * cos(35° ) ; ; ; ; ; ; c**2 -27.032c +90 =0 ; ; a=1; b=-27.032; c=90 ; ; D = b**2 - 4ac = 27.032**2 - 4 * 1 * 90 = 370.729968041 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 27.03 ± sqrt{ 370.73 } }{ 2 } ; ; c_{1,2} = 13.51600873 ± 9.62717466395 ; ; c_{1} = 23.143183394 ; ; c_{2} = 3.88883406605 ; ; ; ;
(c -23.143183394) (c -3.88883406605) = 0 ; ; ; ; c > 0 ; ; ; ; c = 23.143 ; ;


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 = 13.5 ; ; b = 16.5 ; ; c = 3.89 ; ;

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

p = a+b+c = 13.5+16.5+3.89 = 33.89 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.89 }{ 2 } = 16.94 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.94 * (16.94-13.5)(16.94-16.5)(16.94-3.89) } ; ; T = sqrt{ 338.63 } = 18.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.4 }{ 13.5 } = 2.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.4 }{ 16.5 } = 2.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.4 }{ 3.89 } = 9.46 ; ;

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{ 13.5**2-16.5**2-3.89**2 }{ 2 * 16.5 * 3.89 } ) = 35° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.5**2-13.5**2-3.89**2 }{ 2 * 13.5 * 3.89 } ) = 135° 29'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.89**2-13.5**2-16.5**2 }{ 2 * 16.5 * 13.5 } ) = 9° 30'37" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.4 }{ 16.94 } = 1.09 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13.5 }{ 2 * sin 35° } = 11.77 ; ;





#2 Obtuse scalene triangle.

Sides: a = 13.5   b = 16.5   c = 23.14331833947

Area: T = 109.5143673423
Perimeter: p = 53.14331833947
Semiperimeter: s = 26.57215916974

Angle ∠ A = α = 35° = 0.61108652382 rad
Angle ∠ B = β = 44.51103316092° = 44°30'37″ = 0.77768518377 rad
Angle ∠ C = γ = 100.4989668391° = 100°29'23″ = 1.75438755777 rad

Height: ha = 16.22442479146
Height: hb = 13.27443846574
Height: hc = 9.46440111998

Median: ma = 18.93105564847
Median: mb = 17.05547931333
Median: mc = 9.66216906176

Inradius: r = 4.12114570309
Circumradius: R = 11.76882658704

Vertex coordinates: A[23.14331833947; 0] B[0; 0] C[9.6277174664; 9.46440111998]
Centroid: CG[10.92334526862; 3.15546703999]
Coordinates of the circumscribed circle: U[11.57215916974; -2.14325095535]
Coordinates of the inscribed circle: I[10.07215916974; 4.12114570309]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145° = 0.61108652382 rad
∠ B' = β' = 135.4989668391° = 135°29'23″ = 0.77768518377 rad
∠ C' = γ' = 79.51103316092° = 79°30'37″ = 1.75438755777 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 13.5 ; ; b = 16.5 ; ; alpha = 35° ; ; : 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 ; ; ; ; 13.5**2 = 16.5**2 + c**2 - 2 * 16.5 * c * cos(35° ) ; ; ; ; ; ; c**2 -27.032c +90 =0 ; ; a=1; b=-27.032; c=90 ; ; D = b**2 - 4ac = 27.032**2 - 4 * 1 * 90 = 370.729968041 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 27.03 ± sqrt{ 370.73 } }{ 2 } ; ; c_{1,2} = 13.51600873 ± 9.62717466395 ; ; c_{1} = 23.143183394 ; ; c_{2} = 3.88883406605 ; ; ; ; : Nr. 1
(c -23.143183394) (c -3.88883406605) = 0 ; ; ; ; c > 0 ; ; ; ; c = 23.143 ; ; : 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 = 13.5 ; ; b = 16.5 ; ; c = 23.14 ; ;

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

p = a+b+c = 13.5+16.5+23.14 = 53.14 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53.14 }{ 2 } = 26.57 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.57 * (26.57-13.5)(26.57-16.5)(26.57-23.14) } ; ; T = sqrt{ 11993.24 } = 109.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.51 }{ 13.5 } = 16.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.51 }{ 16.5 } = 13.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.51 }{ 23.14 } = 9.46 ; ;

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{ 13.5**2-16.5**2-23.14**2 }{ 2 * 16.5 * 23.14 } ) = 35° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.5**2-13.5**2-23.14**2 }{ 2 * 13.5 * 23.14 } ) = 44° 30'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23.14**2-13.5**2-16.5**2 }{ 2 * 16.5 * 13.5 } ) = 100° 29'23" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.51 }{ 26.57 } = 4.12 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13.5 }{ 2 * sin 35° } = 11.77 ; ; : Nr. 1




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