16 21 23 triangle

Acute scalene triangle.

Sides: a = 16   b = 21   c = 23

Area: T = 162.6655300541
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 42.34326054522° = 42°20'33″ = 0.7399017879 rad
Angle ∠ B = β = 62.13549067337° = 62°8'6″ = 1.08444587029 rad
Angle ∠ C = γ = 75.52224878141° = 75°31'21″ = 1.31881160717 rad

Height: ha = 20.33331625676
Height: hb = 15.49219333848
Height: hc = 14.14548087427

Median: ma = 20.51882845287
Median: mb = 16.88002976164
Median: mc = 14.70554411699

Inradius: r = 5.42221766847
Circumradius: R = 11.87771489284

Vertex coordinates: A[23; 0] B[0; 0] C[7.47882608696; 14.14548087427]
Centroid: CG[10.15994202899; 4.71549362476]
Coordinates of the circumscribed circle: U[11.5; 2.96992872321]
Coordinates of the inscribed circle: I[9; 5.42221766847]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.6577394548° = 137°39'27″ = 0.7399017879 rad
∠ B' = β' = 117.8655093266° = 117°51'54″ = 1.08444587029 rad
∠ C' = γ' = 104.4787512186° = 104°28'39″ = 1.31881160717 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 = 21 ; ; c = 23 ; ;

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

p = a+b+c = 16+21+23 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-16)(30-21)(30-23) } ; ; T = sqrt{ 26460 } = 162.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 162.67 }{ 16 } = 20.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 162.67 }{ 21 } = 15.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 162.67 }{ 23 } = 14.14 ; ;

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**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 42° 20'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-16**2-23**2 }{ 2 * 16 * 23 } ) = 62° 8'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-16**2-21**2 }{ 2 * 21 * 16 } ) = 75° 31'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 162.67 }{ 30 } = 5.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 42° 20'33" } = 11.88 ; ;




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