15 24 27 triangle

Acute scalene triangle.

Sides: a = 15   b = 24   c = 27

Area: T = 179.0987738679
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 33.55773097619° = 33°33'26″ = 0.58656855435 rad
Angle ∠ B = β = 62.18218607153° = 62°10'55″ = 1.08552782045 rad
Angle ∠ C = γ = 84.26108295227° = 84°15'39″ = 1.47106289056 rad

Height: ha = 23.88796984906
Height: hb = 14.92548115566
Height: hc = 13.26664991614

Median: ma = 24.41882308941
Median: mb = 18.24882875909
Median: mc = 14.77332867027

Inradius: r = 5.42772042024
Circumradius: R = 13.5688010506

Vertex coordinates: A[27; 0] B[0; 0] C[7; 13.26664991614]
Centroid: CG[11.33333333333; 4.42221663871]
Coordinates of the circumscribed circle: U[13.5; 1.35768010506]
Coordinates of the inscribed circle: I[9; 5.42772042024]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad
∠ B' = β' = 117.8188139285° = 117°49'5″ = 1.08552782045 rad
∠ C' = γ' = 95.73991704773° = 95°44'21″ = 1.47106289056 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 = 15 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 15+24+27 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-15)(33-24)(33-27) } ; ; T = sqrt{ 32076 } = 179.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 179.1 }{ 15 } = 23.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 179.1 }{ 24 } = 14.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 179.1 }{ 27 } = 13.27 ; ;

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{ 15**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 33° 33'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-15**2-27**2 }{ 2 * 15 * 27 } ) = 62° 10'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-15**2-24**2 }{ 2 * 24 * 15 } ) = 84° 15'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 179.1 }{ 33 } = 5.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 33° 33'26" } = 13.57 ; ;




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