6 12 13 triangle

Acute scalene triangle.

Sides: a = 6   b = 12   c = 13

Area: T = 35.89548116028
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 27.39993615864° = 27°23'58″ = 0.47882090726 rad
Angle ∠ B = β = 66.98216671563° = 66°58'54″ = 1.16990506304 rad
Angle ∠ C = γ = 85.61989712574° = 85°37'8″ = 1.49443329506 rad

Height: ha = 11.96549372009
Height: hb = 5.98224686005
Height: hc = 5.52222787081

Median: ma = 12.14549578015
Median: mb = 8.15547532152
Median: mc = 6.91101374805

Inradius: r = 2.3165794297
Circumradius: R = 6.51990480059

Vertex coordinates: A[13; 0] B[0; 0] C[2.34661538462; 5.52222787081]
Centroid: CG[5.11553846154; 1.84107595694]
Coordinates of the circumscribed circle: U[6.5; 0.49879828338]
Coordinates of the inscribed circle: I[3.5; 2.3165794297]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.6010638414° = 152°36'2″ = 0.47882090726 rad
∠ B' = β' = 113.0188332844° = 113°1'6″ = 1.16990506304 rad
∠ C' = γ' = 94.38110287426° = 94°22'52″ = 1.49443329506 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 = 12 ; ; c = 13 ; ;

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

p = a+b+c = 6+12+13 = 31 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31 }{ 2 } = 15.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.5 * (15.5-6)(15.5-12)(15.5-13) } ; ; T = sqrt{ 1288.44 } = 35.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 35.89 }{ 6 } = 11.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 35.89 }{ 12 } = 5.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 35.89 }{ 13 } = 5.52 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12**2+13**2-6**2 }{ 2 * 12 * 13 } ) = 27° 23'58" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6**2+13**2-12**2 }{ 2 * 6 * 13 } ) = 66° 58'54" ; ; gamma = 180° - alpha - beta = 180° - 27° 23'58" - 66° 58'54" = 85° 37'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 35.89 }{ 15.5 } = 2.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6 }{ 2 * sin 27° 23'58" } = 6.52 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 13**2 - 6**2 } }{ 2 } = 12.145 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 6**2 - 12**2 } }{ 2 } = 8.155 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 6**2 - 13**2 } }{ 2 } = 6.91 ; ;
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