Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=49; b=41; c=88.0000511955 and a=49; b=41; c=89.99994240538.

#1 Obtuse scalene triangle.

Sides: a = 49   b = 41   c = 88.0000511955

Area: T = 0.64114114599
Perimeter: p = 988.0000511955
Semiperimeter: s = 499.0000255978

Angle ∠ A = α = 179.7765914463° = 179°46'33″ = 3.13876816232 rad
Angle ∠ B = β = 0.18875° = 0°11'15″ = 0.00332724923 rad
Angle ∠ C = γ = 0.03765855372° = 0°2'12″ = 0.00106385381 rad

Height: ha = 0.02661800596
Height: hb = 0.03112883639
Height: hc = 0.16603518388

Median: ma = 16.55000124111
Median: mb = 28.55000071854
Median: mc = 454.9999977246

Inradius: r = 0.0133090023
Circumradius: R = 6264.354974116

Vertex coordinates: A[88.0000511955; 0] B[0; 0] C[498.9997376247; 0.16603518388]
Centroid: CG[198.9999296067; 0.05334506129]
Coordinates of the circumscribed circle: U[44.0000255978; 6264.348846401]
Coordinates of the inscribed circle: I[88.0000255978; 0.0133090023]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 0.22440855372° = 0°13'27″ = 3.13876816232 rad
∠ B' = β' = 179.81325° = 179°48'45″ = 0.00332724923 rad
∠ C' = γ' = 179.9633414463° = 179°57'48″ = 0.00106385381 rad




How did we calculate this triangle?

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

a = 49 ; ; b = 41 ; ; beta = 0.188° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 41**2 = 49**2 + c**2 - 2 * 49 * c * cos 0° 11'15" ; ; ; ; ; ; c**2 -97.999c +720 =0 ; ; p=1; q=-97.999; r=720 ; ; D = q**2 - 4pr = 97.999**2 - 4 * 1 * 720 = 6723.89714915 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 98 ± sqrt{ 6723.9 } }{ 2 } ; ; c_{1,2} = 48.99973762 ± 40.9996864291 ; ; c_{1} = 89.9994240491 ; ; c_{2} = 8.00005119086 ; ; ; ;
 text{ Factored form: } ; ; (c -89.9994240491) (c -8.00005119086) = 0 ; ; ; ; c > 0 ; ; ; ; c = 89.999 ; ;
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 = 49 ; ; b = 41 ; ; c = 8 ; ;

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

p = a+b+c = 49+41+8 = 98 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 98 }{ 2 } = 49 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 49 * (49-49)(49-41)(49-8) } ; ; T = sqrt{ 0.41 } = 0.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.64 }{ 49 } = 0.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.64 }{ 41 } = 0.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.64 }{ 8 } = 0.16 ; ;

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{ 41**2+8**2-49**2 }{ 2 * 41 * 8 } ) = 179° 46'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 49**2+8**2-41**2 }{ 2 * 49 * 8 } ) = 0° 11'15" ; ; gamma = 180° - alpha - beta = 180° - 179° 46'33" - 0° 11'15" = 0° 2'12" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.64 }{ 49 } = 0.01 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 49 }{ 2 * sin 179° 46'33" } = 6264.35 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 41**2+2 * 8**2 - 49**2 } }{ 2 } = 16.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8**2+2 * 49**2 - 41**2 } }{ 2 } = 28.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 41**2+2 * 49**2 - 8**2 } }{ 2 } = 45 ; ;







#2 Obtuse scalene triangle.

Sides: a = 49   b = 41   c = 89.99994240538

Area: T = 7.21657865699
Perimeter: p = 179.9999424054
Semiperimeter: s = 909.9997120269

Angle ∠ A = α = 0.22440855372° = 0°13'27″ = 0.00439110304 rad
Angle ∠ B = β = 0.18875° = 0°11'15″ = 0.00332724923 rad
Angle ∠ C = γ = 179.5888414463° = 179°35'18″ = 3.13444091308 rad

Height: ha = 0.29545219008
Height: hb = 0.35219895888
Height: hc = 0.16603518388

Median: ma = 65.54996043119
Median: mb = 69.54996270854
Median: mc = 4.0033238376

Inradius: r = 0.08801756629
Circumradius: R = 6264.354974116

Vertex coordinates: A[89.99994240538; 0] B[0; 0] C[498.9997376247; 0.16603518388]
Centroid: CG[46.33330538928; 0.05334506129]
Coordinates of the circumscribed circle: U[454.9997120269; -6264.188811217]
Coordinates of the inscribed circle: I[498.9997120269; 0.08801756629]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 179.7765914463° = 179°46'33″ = 0.00439110304 rad
∠ B' = β' = 179.81325° = 179°48'45″ = 0.00332724923 rad
∠ C' = γ' = 0.41215855372° = 0°24'42″ = 3.13444091308 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 49 ; ; b = 41 ; ; beta = 0.188° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 41**2 = 49**2 + c**2 - 2 * 49 * c * cos 0° 11'15" ; ; ; ; ; ; c**2 -97.999c +720 =0 ; ; p=1; q=-97.999; r=720 ; ; D = q**2 - 4pr = 97.999**2 - 4 * 1 * 720 = 6723.89714915 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 98 ± sqrt{ 6723.9 } }{ 2 } ; ; c_{1,2} = 48.99973762 ± 40.9996864291 ; ; c_{1} = 89.9994240491 ; ; c_{2} = 8.00005119086 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (c -89.9994240491) (c -8.00005119086) = 0 ; ; ; ; c > 0 ; ; ; ; c = 89.999 ; ; : 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 = 49 ; ; b = 41 ; ; c = 90 ; ;

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

p = a+b+c = 49+41+90 = 180 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 180 }{ 2 } = 90 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90 * (90-49)(90-41)(90-90) } ; ; T = sqrt{ 52.07 } = 7.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.22 }{ 49 } = 0.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.22 }{ 41 } = 0.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.22 }{ 90 } = 0.16 ; ;

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{ 41**2+90**2-49**2 }{ 2 * 41 * 90 } ) = 0° 13'27" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 49**2+90**2-41**2 }{ 2 * 49 * 90 } ) = 0° 11'15" ; ; gamma = 180° - alpha - beta = 180° - 0° 13'27" - 0° 11'15" = 179° 35'18" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.22 }{ 90 } = 0.08 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 49 }{ 2 * sin 0° 13'27" } = 6264.35 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 41**2+2 * 90**2 - 49**2 } }{ 2 } = 65.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 49**2 - 41**2 } }{ 2 } = 69.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 41**2+2 * 49**2 - 90**2 } }{ 2 } = 4.003 ; ;
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