Triangle calculator

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

You have entered side a, b and c.

Acute scalene triangle.

Sides: a = 69   b = 70   c = 41

Area: T = 1360.956554667
Perimeter: p = 180
Semiperimeter: s = 90

Angle ∠ A = α = 71.51439853209° = 71°30'50″ = 1.24881545051 rad
Angle ∠ B = β = 74.18548619223° = 74°11'5″ = 1.29547700957 rad
Angle ∠ C = γ = 34.30111527568° = 34°18'4″ = 0.59986680528 rad

Height: ha = 39.44879868599
Height: hb = 38.88444441904
Height: hc = 66.38880754471

Median: ma = 45.8288484592
Median: mb = 44.67766158074
Median: mc = 66.4109713145

Inradius: r = 15.12217282963
Circumradius: R = 36.37770147536

Vertex coordinates: A[41; 0] B[0; 0] C[18.80548780488; 66.38880754471]
Centroid: CG[19.93549593496; 22.12993584824]
Coordinates of the circumscribed circle: U[20.5; 30.05105774051]
Coordinates of the inscribed circle: I[20; 15.12217282963]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.4866014679° = 108°29'10″ = 1.24881545051 rad
∠ B' = β' = 105.8155138078° = 105°48'55″ = 1.29547700957 rad
∠ C' = γ' = 145.6998847243° = 145°41'56″ = 0.59986680528 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and c.

a = 69 ; ; b = 70 ; ; c = 41 ; ;


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 = 69 ; ; b = 70 ; ; c = 41 ; ; : Nr. 1

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

p = a+b+c = 69+70+41 = 180 ; ;

3. Semiperimeter of the triangle

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

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90 * (90-69)(90-70)(90-41) } ; ; T = sqrt{ 1852200 } = 1360.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1360.96 }{ 69 } = 39.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1360.96 }{ 70 } = 38.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1360.96 }{ 41 } = 66.39 ; ;

6. 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{ 70**2+41**2-69**2 }{ 2 * 70 * 41 } ) = 71° 30'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 69**2+41**2-70**2 }{ 2 * 69 * 41 } ) = 74° 11'5" ; ; gamma = 180° - alpha - beta = 180° - 71° 30'50" - 74° 11'5" = 34° 18'4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1360.96 }{ 90 } = 15.12 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 69 }{ 2 * sin 71° 30'50" } = 36.38 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 70**2+2 * 41**2 - 69**2 } }{ 2 } = 45.828 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 41**2+2 * 69**2 - 70**2 } }{ 2 } = 44.677 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 70**2+2 * 69**2 - 41**2 } }{ 2 } = 66.41 ; ;
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