16 23 24 triangle

Acute scalene triangle.

Sides: a = 16   b = 23   c = 24

Area: T = 176.4255444594
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 39.73438770246° = 39°44'2″ = 0.69334869787 rad
Angle ∠ B = β = 66.76332574659° = 66°45'48″ = 1.16552386621 rad
Angle ∠ C = γ = 73.50328655095° = 73°30'10″ = 1.28328670128 rad

Height: ha = 22.05331805742
Height: hb = 15.34113430081
Height: hc = 14.70221203828

Median: ma = 22.10220361053
Median: mb = 16.84548805279
Median: mc = 15.76438827704

Inradius: r = 5.60108077649
Circumradius: R = 12.51552015634

Vertex coordinates: A[24; 0] B[0; 0] C[6.31325; 14.70221203828]
Centroid: CG[10.10441666667; 4.90107067943]
Coordinates of the circumscribed circle: U[12; 3.55439091396]
Coordinates of the inscribed circle: I[8.5; 5.60108077649]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.2666122975° = 140°15'58″ = 0.69334869787 rad
∠ B' = β' = 113.2376742534° = 113°14'12″ = 1.16552386621 rad
∠ C' = γ' = 106.4977134491° = 106°29'50″ = 1.28328670128 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 = 24 ; ;

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

p = a+b+c = 16+23+24 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-16)(31.5-23)(31.5-24) } ; ; T = sqrt{ 31125.94 } = 176.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 176.43 }{ 16 } = 22.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 176.43 }{ 23 } = 15.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 176.43 }{ 24 } = 14.7 ; ;

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**2-24**2 }{ 2 * 23 * 24 } ) = 39° 44'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-16**2-24**2 }{ 2 * 16 * 24 } ) = 66° 45'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-16**2-23**2 }{ 2 * 23 * 16 } ) = 73° 30'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 176.43 }{ 31.5 } = 5.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 39° 44'2" } = 12.52 ; ;




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