Triangle calculator

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

You have entered side b, c and angle α.

Acute scalene triangle.

Sides: a = 65.94332984162   b = 65   c = 39

Area: T = 1218.39991996
Perimeter: p = 169.9433298416
Semiperimeter: s = 84.97216492081

Angle ∠ A = α = 74° = 1.29215436465 rad
Angle ∠ B = β = 71.35438419583° = 71°21'14″ = 1.24553594761 rad
Angle ∠ C = γ = 34.64661580417° = 34°38'46″ = 0.6054689531 rad

Height: ha = 36.95329346837
Height: hb = 37.48992061416
Height: hc = 62.4822010236

Median: ma = 42.26595592558
Median: mb = 43.34217731871
Median: mc = 62.50220743896

Inradius: r = 14.3398890806
Circumradius: R = 34.33003880706

Vertex coordinates: A[39; 0] B[0; 0] C[21.08435718719; 62.4822010236]
Centroid: CG[20.02878572906; 20.82773367453]
Coordinates of the circumscribed circle: U[19.5; 28.21881966432]
Coordinates of the inscribed circle: I[19.97216492081; 14.3398890806]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 106° = 1.29215436465 rad
∠ B' = β' = 108.6466158042° = 108°38'46″ = 1.24553594761 rad
∠ C' = γ' = 145.3543841958° = 145°21'14″ = 0.6054689531 rad

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How did we calculate this triangle?

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

b = 65 ; ; c = 39 ; ; alpha = 74° ; ;

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{ 65**2+39**2 - 2 * 65 * 39 * cos(74° ) } ; ; a = 65.94 ; ;


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 = 65.94 ; ; b = 65 ; ; c = 39 ; ;

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

p = a+b+c = 65.94+65+39 = 169.94 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 169.94 }{ 2 } = 84.97 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 84.97 * (84.97-65.94)(84.97-65)(84.97-39) } ; ; T = sqrt{ 1484496.61 } = 1218.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1218.4 }{ 65.94 } = 36.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1218.4 }{ 65 } = 37.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1218.4 }{ 39 } = 62.48 ; ;

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{ 65**2+39**2-65.94**2 }{ 2 * 65 * 39 } ) = 74° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 65.94**2+39**2-65**2 }{ 2 * 65.94 * 39 } ) = 71° 21'14" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 65.94**2+65**2-39**2 }{ 2 * 65.94 * 65 } ) = 34° 38'46" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1218.4 }{ 84.97 } = 14.34 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 65.94 }{ 2 * sin 74° } = 34.3 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 65**2+2 * 39**2 - 65.94**2 } }{ 2 } = 42.26 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 39**2+2 * 65.94**2 - 65**2 } }{ 2 } = 43.342 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 65**2+2 * 65.94**2 - 39**2 } }{ 2 } = 62.502 ; ;
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