14 15 16 triangle

Acute scalene triangle.

Sides: a = 14   b = 15   c = 16

Area: T = 96.55879489219
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 53.57664263577° = 53°34'35″ = 0.93550850414 rad
Angle ∠ B = β = 59.55659700416° = 59°33'21″ = 1.03994477664 rad
Angle ∠ C = γ = 66.86876036007° = 66°52'3″ = 1.16770598458 rad

Height: ha = 13.79439927031
Height: hb = 12.87443931896
Height: hc = 12.07697436152

Median: ma = 13.83883525031
Median: mb = 13.02988142208
Median: mc = 12.10437184369

Inradius: r = 4.29114643965
Circumradius: R = 8.69994391387

Vertex coordinates: A[16; 0] B[0; 0] C[7.094375; 12.07697436152]
Centroid: CG[7.69879166667; 4.02332478717]
Coordinates of the circumscribed circle: U[8; 3.41876368045]
Coordinates of the inscribed circle: I[7.5; 4.29114643965]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.4243573642° = 126°25'25″ = 0.93550850414 rad
∠ B' = β' = 120.4444029958° = 120°26'39″ = 1.03994477664 rad
∠ C' = γ' = 113.1322396399° = 113°7'57″ = 1.16770598458 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 = 14 ; ; b = 15 ; ; c = 16 ; ;

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

p = a+b+c = 14+15+16 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-14)(22.5-15)(22.5-16) } ; ; T = sqrt{ 9323.44 } = 96.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 96.56 }{ 14 } = 13.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 96.56 }{ 15 } = 12.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 96.56 }{ 16 } = 12.07 ; ;

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{ 14**2-15**2-16**2 }{ 2 * 15 * 16 } ) = 53° 34'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-14**2-16**2 }{ 2 * 14 * 16 } ) = 59° 33'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-14**2-15**2 }{ 2 * 15 * 14 } ) = 66° 52'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 96.56 }{ 22.5 } = 4.29 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 53° 34'35" } = 8.7 ; ;




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