13 24 26 triangle

Acute scalene triangle.

Sides: a = 13   b = 24   c = 26

Area: T = 155.0433340715
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 29.7977347297° = 29°47'50″ = 0.52200618187 rad
Angle ∠ B = β = 66.55112264175° = 66°33'4″ = 1.16215380222 rad
Angle ∠ C = γ = 83.65114262854° = 83°39'5″ = 1.46599928127 rad

Height: ha = 23.85328216484
Height: hb = 12.92202783929
Height: hc = 11.92664108242

Median: ma = 24.16109188567
Median: mb = 16.68883192683
Median: mc = 14.26553426177

Inradius: r = 4.92220108163
Circumradius: R = 13.08802135109

Vertex coordinates: A[26; 0] B[0; 0] C[5.17330769231; 11.92664108242]
Centroid: CG[10.3911025641; 3.97554702747]
Coordinates of the circumscribed circle: U[13; 1.44663697632]
Coordinates of the inscribed circle: I[7.5; 4.92220108163]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.2032652703° = 150°12'10″ = 0.52200618187 rad
∠ B' = β' = 113.4498773582° = 113°26'56″ = 1.16215380222 rad
∠ C' = γ' = 96.34985737146° = 96°20'55″ = 1.46599928127 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 = 13 ; ; b = 24 ; ; c = 26 ; ;

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

p = a+b+c = 13+24+26 = 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 * (31.5-13)(31.5-24)(31.5-26) } ; ; T = sqrt{ 24038.44 } = 155.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 155.04 }{ 13 } = 23.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 155.04 }{ 24 } = 12.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 155.04 }{ 26 } = 11.93 ; ;

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{ 13**2-24**2-26**2 }{ 2 * 24 * 26 } ) = 29° 47'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-13**2-26**2 }{ 2 * 13 * 26 } ) = 66° 33'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 26**2-13**2-24**2 }{ 2 * 24 * 13 } ) = 83° 39'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 155.04 }{ 31.5 } = 4.92 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 29° 47'50" } = 13.08 ; ;




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