18 20 23 triangle

Acute scalene triangle.

Sides: a = 18   b = 20   c = 23

Area: T = 173.2732725782
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 48.88222476784° = 48°52'56″ = 0.85331561678 rad
Angle ∠ B = β = 56.83216133696° = 56°49'54″ = 0.99218987725 rad
Angle ∠ C = γ = 74.2866138952° = 74°17'10″ = 1.29765377133 rad

Height: ha = 19.25325250869
Height: hb = 17.32772725782
Height: hc = 15.06771935463

Median: ma = 19.58331560276
Median: mb = 18.06993109996
Median: mc = 15.15875063912

Inradius: r = 5.68110729765
Circumradius: R = 11.94664848877

Vertex coordinates: A[23; 0] B[0; 0] C[9.8487826087; 15.06771935463]
Centroid: CG[10.94992753623; 5.02223978488]
Coordinates of the circumscribed circle: U[11.5; 3.23655063237]
Coordinates of the inscribed circle: I[10.5; 5.68110729765]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.1187752322° = 131°7'4″ = 0.85331561678 rad
∠ B' = β' = 123.168838663° = 123°10'6″ = 0.99218987725 rad
∠ C' = γ' = 105.7143861048° = 105°42'50″ = 1.29765377133 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 = 18 ; ; b = 20 ; ; c = 23 ; ;

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

p = a+b+c = 18+20+23 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-18)(30.5-20)(30.5-23) } ; ; T = sqrt{ 30023.44 } = 173.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 173.27 }{ 18 } = 19.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 173.27 }{ 20 } = 17.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 173.27 }{ 23 } = 15.07 ; ;

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{ 18**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 48° 52'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-18**2-23**2 }{ 2 * 18 * 23 } ) = 56° 49'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-18**2-20**2 }{ 2 * 20 * 18 } ) = 74° 17'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 173.27 }{ 30.5 } = 5.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 48° 52'56" } = 11.95 ; ;




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