23 26 27 triangle

Acute scalene triangle.

Sides: a = 23   b = 26   c = 27

Area: T = 274.2999106816
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 51.39661856156° = 51°23'46″ = 0.8977032662 rad
Angle ∠ B = β = 62.05766048416° = 62°3'24″ = 1.08330920771 rad
Angle ∠ C = γ = 66.54772095428° = 66°32'50″ = 1.16114679145 rad

Height: ha = 23.85220962449
Height: hb = 21.10999312935
Height: hc = 20.31884523567

Median: ma = 23.88799078725
Median: mb = 21.44876105895
Median: mc = 20.5

Inradius: r = 7.21883975478
Circumradius: R = 14.71656877281

Vertex coordinates: A[27; 0] B[0; 0] C[10.77877777778; 20.31884523567]
Centroid: CG[12.59325925926; 6.77328174522]
Coordinates of the circumscribed circle: U[13.5; 5.85767452831]
Coordinates of the inscribed circle: I[12; 7.21883975478]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.6043814384° = 128°36'14″ = 0.8977032662 rad
∠ B' = β' = 117.9433395158° = 117°56'36″ = 1.08330920771 rad
∠ C' = γ' = 113.4532790457° = 113°27'10″ = 1.16114679145 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 = 23 ; ; b = 26 ; ; c = 27 ; ;

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

p = a+b+c = 23+26+27 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-23)(38-26)(38-27) } ; ; T = sqrt{ 75240 } = 274.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 274.3 }{ 23 } = 23.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 274.3 }{ 26 } = 21.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 274.3 }{ 27 } = 20.32 ; ;

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{ 23**2-26**2-27**2 }{ 2 * 26 * 27 } ) = 51° 23'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 62° 3'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-23**2-26**2 }{ 2 * 26 * 23 } ) = 66° 32'50" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 274.3 }{ 38 } = 7.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 51° 23'46" } = 14.72 ; ;




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