4 12 13 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 12   c = 13

Area: T = 23.89442984831
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 17.83986287523° = 17°50'19″ = 0.31113428058 rad
Angle ∠ B = β = 66.78219922566° = 66°46'55″ = 1.16655656459 rad
Angle ∠ C = γ = 95.37993789911° = 95°22'46″ = 1.66546842019 rad

Height: ha = 11.94771492416
Height: hb = 3.98223830805
Height: hc = 3.67660459205

Median: ma = 12.34990890352
Median: mb = 7.51766481892
Median: mc = 6.14441028637

Inradius: r = 1.6487882654
Circumradius: R = 6.5298754134

Vertex coordinates: A[13; 0] B[0; 0] C[1.57769230769; 3.67660459205]
Centroid: CG[4.8598974359; 1.22553486402]
Coordinates of the circumscribed circle: U[6.5; -0.61220707001]
Coordinates of the inscribed circle: I[2.5; 1.6487882654]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.1611371248° = 162°9'41″ = 0.31113428058 rad
∠ B' = β' = 113.2188007743° = 113°13'5″ = 1.16655656459 rad
∠ C' = γ' = 84.62106210089° = 84°37'14″ = 1.66546842019 rad

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How did we calculate this triangle?

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 = 4 ; ; b = 12 ; ; c = 13 ; ;

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

p = a+b+c = 4+12+13 = 29 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29 }{ 2 } = 14.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.5 * (14.5-4)(14.5-12)(14.5-13) } ; ; T = sqrt{ 570.94 } = 23.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23.89 }{ 4 } = 11.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23.89 }{ 12 } = 3.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23.89 }{ 13 } = 3.68 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 4**2-12**2-13**2 }{ 2 * 12 * 13 } ) = 17° 50'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-4**2-13**2 }{ 2 * 4 * 13 } ) = 66° 46'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-4**2-12**2 }{ 2 * 12 * 4 } ) = 95° 22'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23.89 }{ 14.5 } = 1.65 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 17° 50'19" } = 6.53 ; ;




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