Triangle calculator

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

You have entered side b, c and angle β.

Triangle has two solutions: a=7.9099114914; b=33; c=38 and a=44.88549212409; b=33; c=38.

#1 Obtuse scalene triangle.

Sides: a = 7.9099114914   b = 33   c = 38

Area: T = 108.0977481719
Perimeter: p = 78.9099114914
Semiperimeter: s = 39.4554557457

Angle ∠ A = α = 9.92876383247° = 9°55'40″ = 0.17332699757 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 124.0722361675° = 124°4'20″ = 2.1655471222 rad

Height: ha = 27.33549124129
Height: hb = 6.55113625284
Height: hc = 5.68993411431

Median: ma = 35.36875200618
Median: mb = 21.93223288632
Median: mc = 14.6555273773

Inradius: r = 2.74397970903
Circumradius: R = 22.93876992518

Vertex coordinates: A[38; 0] B[0; 0] C[5.49441328779; 5.68993411431]
Centroid: CG[14.49880442926; 1.89664470477]
Coordinates of the circumscribed circle: U[19; -12.8510604926]
Coordinates of the inscribed circle: I[6.4554557457; 2.74397970903]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.0722361675° = 170°4'20″ = 0.17332699757 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 55.92876383247° = 55°55'40″ = 2.1655471222 rad




How did we calculate this triangle?

1. Input data entered: side b, c and angle β.

b = 33 ; ; c = 38 ; ; beta = 46° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 33**2 = 38**2 + a**2 - 2 * 38 * a * cos 46° ; ; ; ; ; ; a**2 -52.794a +355 =0 ; ; p=1; q=-52.794; r=355 ; ; D = q**2 - 4pr = 52.794**2 - 4 * 1 * 355 = 1367.21025352 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 52.79 ± sqrt{ 1367.21 } }{ 2 } ; ; a_{1,2} = 26.39701808 ± 18.4879031634 ; ; a_{1} = 44.8849212434 ; ; a_{2} = 7.90911491656 ; ; ; ; text{ Factored form: } ; ; (a -44.8849212434) (a -7.90911491656) = 0 ; ; ; ; a > 0 ; ; ; ; a = 44.885 ; ;


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 = 7.91 ; ; b = 33 ; ; c = 38 ; ;

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

p = a+b+c = 7.91+33+38 = 78.91 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78.91 }{ 2 } = 39.45 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.45 * (39.45-7.91)(39.45-33)(39.45-38) } ; ; T = sqrt{ 11685.07 } = 108.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 108.1 }{ 7.91 } = 27.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 108.1 }{ 33 } = 6.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 108.1 }{ 38 } = 5.69 ; ;

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{ 33**2+38**2-7.91**2 }{ 2 * 33 * 38 } ) = 9° 55'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.91**2+38**2-33**2 }{ 2 * 7.91 * 38 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 9° 55'40" - 46° = 124° 4'20" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 108.1 }{ 39.45 } = 2.74 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.91 }{ 2 * sin 9° 55'40" } = 22.94 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 38**2 - 7.91**2 } }{ 2 } = 35.368 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 38**2+2 * 7.91**2 - 33**2 } }{ 2 } = 21.932 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 7.91**2 - 38**2 } }{ 2 } = 14.655 ; ;







#2 Acute scalene triangle.

Sides: a = 44.88549212409   b = 33   c = 38

Area: T = 613.4632695389
Perimeter: p = 115.8854921241
Semiperimeter: s = 57.94224606204

Angle ∠ A = α = 78.07223616753° = 78°4'20″ = 1.3632619766 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 55.92876383247° = 55°55'40″ = 0.97661214316 rad

Height: ha = 27.33549124129
Height: hb = 37.18795572963
Height: hc = 32.28875102836

Median: ma = 27.61994851744
Median: mb = 38.17216920951
Median: mc = 34.50883769163

Inradius: r = 10.58774463877
Circumradius: R = 22.93876992518

Vertex coordinates: A[38; 0] B[0; 0] C[31.18796862474; 32.28875102836]
Centroid: CG[23.06598954158; 10.76325034279]
Coordinates of the circumscribed circle: U[19; 12.8510604926]
Coordinates of the inscribed circle: I[24.94224606204; 10.58774463877]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 101.9287638325° = 101°55'40″ = 1.3632619766 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 124.0722361675° = 124°4'20″ = 0.97661214316 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side b, c and angle β.

b = 33 ; ; c = 38 ; ; beta = 46° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 33**2 = 38**2 + a**2 - 2 * 38 * a * cos 46° ; ; ; ; ; ; a**2 -52.794a +355 =0 ; ; p=1; q=-52.794; r=355 ; ; D = q**2 - 4pr = 52.794**2 - 4 * 1 * 355 = 1367.21025352 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 52.79 ± sqrt{ 1367.21 } }{ 2 } ; ; a_{1,2} = 26.39701808 ± 18.4879031634 ; ; a_{1} = 44.8849212434 ; ; a_{2} = 7.90911491656 ; ; ; ; text{ Factored form: } ; ; (a -44.8849212434) (a -7.90911491656) = 0 ; ; ; ; a > 0 ; ; ; ; a = 44.885 ; ; : 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 = 44.88 ; ; b = 33 ; ; c = 38 ; ;

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

p = a+b+c = 44.88+33+38 = 115.88 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 115.88 }{ 2 } = 57.94 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.94 * (57.94-44.88)(57.94-33)(57.94-38) } ; ; T = sqrt{ 376336.48 } = 613.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 613.46 }{ 44.88 } = 27.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 613.46 }{ 33 } = 37.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 613.46 }{ 38 } = 32.29 ; ;

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{ 33**2+38**2-44.88**2 }{ 2 * 33 * 38 } ) = 78° 4'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 44.88**2+38**2-33**2 }{ 2 * 44.88 * 38 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 78° 4'20" - 46° = 55° 55'40" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 613.46 }{ 57.94 } = 10.59 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 44.88 }{ 2 * sin 78° 4'20" } = 22.94 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 38**2 - 44.88**2 } }{ 2 } = 27.619 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 38**2+2 * 44.88**2 - 33**2 } }{ 2 } = 38.172 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 44.88**2 - 38**2 } }{ 2 } = 34.508 ; ;
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