10 15 16 triangle

Acute scalene triangle.

Sides: a = 10   b = 15   c = 16

Area: T = 72.98992971606
Perimeter: p = 41
Semiperimeter: s = 20.5

Angle ∠ A = α = 37.46326510725° = 37°27'46″ = 0.65438466077 rad
Angle ∠ B = β = 65.83444206747° = 65°50'4″ = 1.14990274019 rad
Angle ∠ C = γ = 76.70329282528° = 76°42'11″ = 1.33987186439 rad

Height: ha = 14.59878594321
Height: hb = 9.73219062881
Height: hc = 9.12436621451

Median: ma = 14.68799182559
Median: mb = 11.03440382454
Median: mc = 9.92547166206

Inradius: r = 3.566045352
Circumradius: R = 8.22203833074

Vertex coordinates: A[16; 0] B[0; 0] C[4.094375; 9.12436621451]
Centroid: CG[6.69879166667; 3.0411220715]
Coordinates of the circumscribed circle: U[8; 1.89106881607]
Coordinates of the inscribed circle: I[5.5; 3.566045352]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.5377348927° = 142°32'14″ = 0.65438466077 rad
∠ B' = β' = 114.1665579325° = 114°9'56″ = 1.14990274019 rad
∠ C' = γ' = 103.2977071747° = 103°17'49″ = 1.33987186439 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 = 10 ; ; b = 15 ; ; c = 16 ; ;

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

p = a+b+c = 10+15+16 = 41 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41 }{ 2 } = 20.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.5 * (20.5-10)(20.5-15)(20.5-16) } ; ; T = sqrt{ 5327.44 } = 72.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 72.99 }{ 10 } = 14.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 72.99 }{ 15 } = 9.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 72.99 }{ 16 } = 9.12 ; ;

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{ 10**2-15**2-16**2 }{ 2 * 15 * 16 } ) = 37° 27'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-10**2-16**2 }{ 2 * 10 * 16 } ) = 65° 50'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-10**2-15**2 }{ 2 * 15 * 10 } ) = 76° 42'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 72.99 }{ 20.5 } = 3.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 37° 27'46" } = 8.22 ; ;




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