18 21 29 triangle

Obtuse scalene triangle.

Sides: a = 18   b = 21   c = 29

Area: T = 188.0432548377
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 38.13771011066° = 38°8'14″ = 0.66656179815 rad
Angle ∠ B = β = 46.09332596507° = 46°5'36″ = 0.80444791439 rad
Angle ∠ C = γ = 95.77696392427° = 95°46'11″ = 1.67114955282 rad

Height: ha = 20.89436164863
Height: hb = 17.90988141311
Height: hc = 12.96884516122

Median: ma = 23.66443191324
Median: mb = 21.73113138121
Median: mc = 13.12444047484

Inradius: r = 5.53106631875
Circumradius: R = 14.57438292937

Vertex coordinates: A[29; 0] B[0; 0] C[12.48327586207; 12.96884516122]
Centroid: CG[13.82875862069; 4.32328172041]
Coordinates of the circumscribed circle: U[14.5; -1.46550939502]
Coordinates of the inscribed circle: I[13; 5.53106631875]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.8632898893° = 141°51'46″ = 0.66656179815 rad
∠ B' = β' = 133.9076740349° = 133°54'24″ = 0.80444791439 rad
∠ C' = γ' = 84.23303607573° = 84°13'49″ = 1.67114955282 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 = 21 ; ; c = 29 ; ;

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

p = a+b+c = 18+21+29 = 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)(34-21)(34-29) } ; ; T = sqrt{ 35360 } = 188.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 188.04 }{ 18 } = 20.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 188.04 }{ 21 } = 17.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 188.04 }{ 29 } = 12.97 ; ;

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**2-21**2-29**2 }{ 2 * 21 * 29 } ) = 38° 8'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-18**2-29**2 }{ 2 * 18 * 29 } ) = 46° 5'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-18**2-21**2 }{ 2 * 21 * 18 } ) = 95° 46'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 188.04 }{ 34 } = 5.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 38° 8'14" } = 14.57 ; ;




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