Triangle calculator

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

You have entered side b, c and angle α.

Obtuse scalene triangle.

Sides: a = 1.85547031015   b = 90   c = 89

Area: T = 69.89768877813
Perimeter: p = 180.8554703102
Semiperimeter: s = 90.42773515507

Angle ∠ A = α = 1° = 0.01774532925 rad
Angle ∠ B = β = 122.1265748392° = 122°7'33″ = 2.13114964109 rad
Angle ∠ C = γ = 56.87442516077° = 56°52'27″ = 0.99326429502 rad

Height: ha = 75.37325895275
Height: hb = 1.55332641729
Height: hc = 1.57107165794

Median: ma = 89.49765922206
Median: mb = 44.0143861019
Median: mc = 45.51334041992

Inradius: r = 0.77329617929
Circumradius: R = 53.13660276343

Vertex coordinates: A[89; 0] B[0; 0] C[-0.98662925641; 1.57107165794]
Centroid: CG[29.33879024786; 0.52435721931]
Coordinates of the circumscribed circle: U[44.5; 29.0387689866]
Coordinates of the inscribed circle: I[0.42773515507; 0.77329617929]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 179° = 0.01774532925 rad
∠ B' = β' = 57.87442516077° = 57°52'27″ = 2.13114964109 rad
∠ C' = γ' = 123.1265748392° = 123°7'33″ = 0.99326429502 rad

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How did we calculate this triangle?

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

b = 90 ; ; c = 89 ; ; alpha = 1° ; ;

2. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 90**2+89**2 - 2 * 90 * 89 * cos 1° } ; ; a = 1.85 ; ;
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 = 1.85 ; ; b = 90 ; ; c = 89 ; ;

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

p = a+b+c = 1.85+90+89 = 180.85 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 180.85 }{ 2 } = 90.43 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90.43 * (90.43-1.85)(90.43-90)(90.43-89) } ; ; T = sqrt{ 4885.57 } = 69.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.9 }{ 1.85 } = 75.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.9 }{ 90 } = 1.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.9 }{ 89 } = 1.57 ; ;

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{ 90**2+89**2-1.85**2 }{ 2 * 90 * 89 } ) = 1° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.85**2+89**2-90**2 }{ 2 * 1.85 * 89 } ) = 122° 7'33" ; ;
 gamma = 180° - alpha - beta = 180° - 1° - 122° 7'33" = 56° 52'27" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.9 }{ 90.43 } = 0.77 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.85 }{ 2 * sin 1° } = 53.14 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 89**2 - 1.85**2 } }{ 2 } = 89.497 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 89**2+2 * 1.85**2 - 90**2 } }{ 2 } = 44.014 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 1.85**2 - 89**2 } }{ 2 } = 45.513 ; ;
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