13 16 18 triangle

Acute scalene triangle.

Sides: a = 13   b = 16   c = 18

Area: T = 100.8888242625
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 44.47661894092° = 44°28'34″ = 0.77662559439 rad
Angle ∠ B = β = 59.57549868512° = 59°34'30″ = 1.04397796724 rad
Angle ∠ C = γ = 75.94988237396° = 75°56'56″ = 1.32655570373 rad

Height: ha = 15.52112680962
Height: hb = 12.61110303281
Height: hc = 11.21098047361

Median: ma = 15.74400762387
Median: mb = 13.50992560861
Median: mc = 11.46773449412

Inradius: r = 4.29331167075
Circumradius: R = 9.27875924691

Vertex coordinates: A[18; 0] B[0; 0] C[6.58333333333; 11.21098047361]
Centroid: CG[8.19444444444; 3.73766015787]
Coordinates of the circumscribed circle: U[9; 2.25224924024]
Coordinates of the inscribed circle: I[7.5; 4.29331167075]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5243810591° = 135°31'26″ = 0.77662559439 rad
∠ B' = β' = 120.4255013149° = 120°25'30″ = 1.04397796724 rad
∠ C' = γ' = 104.051117626° = 104°3'4″ = 1.32655570373 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 = 13 ; ; b = 16 ; ; c = 18 ; ;

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

p = a+b+c = 13+16+18 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-13)(23.5-16)(23.5-18) } ; ; T = sqrt{ 10178.44 } = 100.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 100.89 }{ 13 } = 15.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 100.89 }{ 16 } = 12.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 100.89 }{ 18 } = 11.21 ; ;

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{ 13**2-16**2-18**2 }{ 2 * 16 * 18 } ) = 44° 28'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-13**2-18**2 }{ 2 * 13 * 18 } ) = 59° 34'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18**2-13**2-16**2 }{ 2 * 16 * 13 } ) = 75° 56'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 100.89 }{ 23.5 } = 4.29 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 44° 28'34" } = 9.28 ; ;




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