21 21 29 triangle

Acute isosceles triangle.

Sides: a = 21   b = 21   c = 29

Area: T = 220.2621634199
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 46.33221847017° = 46°19'56″ = 0.80986491727 rad
Angle ∠ B = β = 46.33221847017° = 46°19'56″ = 0.80986491727 rad
Angle ∠ C = γ = 87.33656305966° = 87°20'8″ = 1.52442943082 rad

Height: ha = 20.97772984951
Height: hb = 20.97772984951
Height: hc = 15.1990457531

Median: ma = 23.0388012067
Median: mb = 23.0388012067
Median: mc = 15.1990457531

Inradius: r = 6.2054553076
Circumradius: R = 14.51656918118

Vertex coordinates: A[29; 0] B[0; 0] C[14.5; 15.1990457531]
Centroid: CG[14.5; 5.06334858437]
Coordinates of the circumscribed circle: U[14.5; 0.67547657191]
Coordinates of the inscribed circle: I[14.5; 6.2054553076]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.6687815298° = 133°40'4″ = 0.80986491727 rad
∠ B' = β' = 133.6687815298° = 133°40'4″ = 0.80986491727 rad
∠ C' = γ' = 92.66443694034° = 92°39'52″ = 1.52442943082 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 = 21 ; ; b = 21 ; ; c = 29 ; ;

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

p = a+b+c = 21+21+29 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-21)(35.5-21)(35.5-29) } ; ; T = sqrt{ 48515.19 } = 220.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 220.26 }{ 21 } = 20.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 220.26 }{ 21 } = 20.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 220.26 }{ 29 } = 15.19 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 21**2+29**2-21**2 }{ 2 * 21 * 29 } ) = 46° 19'56" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 21**2+29**2-21**2 }{ 2 * 21 * 29 } ) = 46° 19'56" ; ; gamma = 180° - alpha - beta = 180° - 46° 19'56" - 46° 19'56" = 87° 20'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 220.26 }{ 35.5 } = 6.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 46° 19'56" } = 14.52 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 29**2 - 21**2 } }{ 2 } = 23.038 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29**2+2 * 21**2 - 21**2 } }{ 2 } = 23.038 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 21**2 - 29**2 } }{ 2 } = 15.19 ; ;
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