16 23 28 triangle

Acute scalene triangle.

Sides: a = 16   b = 23   c = 28

Area: T = 1843.999830163
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 34.85498677543° = 34°51' = 0.60882449362 rad
Angle ∠ B = β = 55.22879797962° = 55°13'41″ = 0.96439100867 rad
Angle ∠ C = γ = 89.92221524495° = 89°55'20″ = 1.56994376307 rad

Height: ha = 232.9999787704
Height: hb = 165.9999852316
Height: hc = 13.14328450116

Median: ma = 24.34113228893
Median: mb = 19.69113686675
Median: mc = 14.01878457689

Inradius: r = 5.49325322437
Circumradius: R = 144.0000129224

Vertex coordinates: A[28; 0] B[0; 0] C[9.125; 13.14328450116]
Centroid: CG[12.375; 4.38109483372]
Coordinates of the circumscribed circle: U[14; 0.01990217567]
Coordinates of the inscribed circle: I[10.5; 5.49325322437]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.1550132246° = 145°9' = 0.60882449362 rad
∠ B' = β' = 124.7722020204° = 124°46'19″ = 0.96439100867 rad
∠ C' = γ' = 90.07878475505° = 90°4'40″ = 1.56994376307 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 = 16 ; ; b = 23 ; ; c = 28 ; ;

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

p = a+b+c = 16+23+28 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-16)(33.5-23)(33.5-28) } ; ; T = sqrt{ 33855.94 } = 184 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 184 }{ 16 } = 23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 184 }{ 23 } = 16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 184 }{ 28 } = 13.14 ; ;

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{ 16**2-23**2-28**2 }{ 2 * 23 * 28 } ) = 34° 51' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-16**2-28**2 }{ 2 * 16 * 28 } ) = 55° 13'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-16**2-23**2 }{ 2 * 23 * 16 } ) = 89° 55'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 184 }{ 33.5 } = 5.49 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 34° 51' } = 14 ; ;




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