18 23 27 triangle

Acute scalene triangle.

Sides: a = 18   b = 23   c = 27

Area: T = 204.6665580887
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 41.23549578936° = 41°14'6″ = 0.72196857822 rad
Angle ∠ B = β = 57.37879709914° = 57°22'41″ = 1.00114345119 rad
Angle ∠ C = γ = 81.3877071115° = 81°23'13″ = 1.42204723595 rad

Height: ha = 22.74106200986
Height: hb = 17.79770070337
Height: hc = 15.16604133991

Median: ma = 23.40993998214
Median: mb = 19.85657296517
Median: mc = 15.62884996081

Inradius: r = 6.02195759085
Circumradius: R = 13.65439812307

Vertex coordinates: A[27; 0] B[0; 0] C[9.70437037037; 15.16604133991]
Centroid: CG[12.23545679012; 5.0533471133]
Coordinates of the circumscribed circle: U[13.5; 2.04547991215]
Coordinates of the inscribed circle: I[11; 6.02195759085]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.7655042106° = 138°45'54″ = 0.72196857822 rad
∠ B' = β' = 122.6222029009° = 122°37'19″ = 1.00114345119 rad
∠ C' = γ' = 98.6132928885° = 98°36'47″ = 1.42204723595 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 = 23 ; ; c = 27 ; ;

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

p = a+b+c = 18+23+27 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-18)(34-23)(34-27) } ; ; T = sqrt{ 41888 } = 204.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 204.67 }{ 18 } = 22.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 204.67 }{ 23 } = 17.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 204.67 }{ 27 } = 15.16 ; ;

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**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 41° 14'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-18**2-27**2 }{ 2 * 18 * 27 } ) = 57° 22'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-18**2-23**2 }{ 2 * 23 * 18 } ) = 81° 23'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 204.67 }{ 34 } = 6.02 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 41° 14'6" } = 13.65 ; ;




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