6 10 13 triangle

Obtuse scalene triangle.

Sides: a = 6   b = 10   c = 13

Area: T = 28.84333267845
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 26.34329755443° = 26°20'35″ = 0.4659771658 rad
Angle ∠ B = β = 47.69550102928° = 47°41'42″ = 0.83224349664 rad
Angle ∠ C = γ = 105.9622014163° = 105°57'43″ = 1.84993860292 rad

Height: ha = 9.61444422615
Height: hb = 5.76986653569
Height: hc = 4.43774348899

Median: ma = 11.20326782512
Median: mb = 8.80334084308
Median: mc = 5.07444457825

Inradius: r = 1.98991949507
Circumradius: R = 6.76106625774

Vertex coordinates: A[13; 0] B[0; 0] C[4.03884615385; 4.43774348899]
Centroid: CG[5.67994871795; 1.47991449633]
Coordinates of the circumscribed circle: U[6.5; -1.85991822088]
Coordinates of the inscribed circle: I[4.5; 1.98991949507]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.6577024456° = 153°39'25″ = 0.4659771658 rad
∠ B' = β' = 132.3054989707° = 132°18'18″ = 0.83224349664 rad
∠ C' = γ' = 74.03879858372° = 74°2'17″ = 1.84993860292 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 = 6 ; ; b = 10 ; ; c = 13 ; ;

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

p = a+b+c = 6+10+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 * (14.5-6)(14.5-10)(14.5-13) } ; ; T = sqrt{ 831.94 } = 28.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 28.84 }{ 6 } = 9.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 28.84 }{ 10 } = 5.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 28.84 }{ 13 } = 4.44 ; ;

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{ 6**2-10**2-13**2 }{ 2 * 10 * 13 } ) = 26° 20'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-6**2-13**2 }{ 2 * 6 * 13 } ) = 47° 41'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-6**2-10**2 }{ 2 * 10 * 6 } ) = 105° 57'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 28.84 }{ 14.5 } = 1.99 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 26° 20'35" } = 6.76 ; ;




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