7 11 13 triangle

Acute scalene triangle.

Sides: a = 7   b = 11   c = 13

Area: T = 38.49991883031
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 32.57881984923° = 32°34'42″ = 0.56985968281 rad
Angle ∠ B = β = 57.79438546387° = 57°47'38″ = 1.00986930509 rad
Angle ∠ C = γ = 89.6287946869° = 89°37'41″ = 1.56443027747 rad

Height: ha = 110.9997680866
Height: hb = 76.9998524188
Height: hc = 5.92329520466

Median: ma = 11.52217186218
Median: mb = 8.87441196746
Median: mc = 6.53883484153

Inradius: r = 2.48438186002
Circumradius: R = 6.55001370426

Vertex coordinates: A[13; 0] B[0; 0] C[3.73107692308; 5.92329520466]
Centroid: CG[5.57769230769; 1.97443173489]
Coordinates of the circumscribed circle: U[6.5; 0.04222086821]
Coordinates of the inscribed circle: I[4.5; 2.48438186002]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.4221801508° = 147°25'18″ = 0.56985968281 rad
∠ B' = β' = 122.2066145361° = 122°12'22″ = 1.00986930509 rad
∠ C' = γ' = 90.3722053131° = 90°22'19″ = 1.56443027747 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 = 7 ; ; b = 11 ; ; c = 13 ; ;

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

p = a+b+c = 7+11+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 * (15.5-7)(15.5-11)(15.5-13) } ; ; T = sqrt{ 1482.19 } = 38.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 38.5 }{ 7 } = 11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 38.5 }{ 11 } = 7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 38.5 }{ 13 } = 5.92 ; ;

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{ 7**2-11**2-13**2 }{ 2 * 11 * 13 } ) = 32° 34'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-7**2-13**2 }{ 2 * 7 * 13 } ) = 57° 47'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-7**2-11**2 }{ 2 * 11 * 7 } ) = 89° 37'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 38.5 }{ 15.5 } = 2.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 32° 34'42" } = 6.5 ; ;




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