21 21 24 triangle

Acute isosceles triangle.

Sides: a = 21   b = 21   c = 24

Area: T = 206.8044255275
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 55.1550095421° = 55°9' = 0.96325507479 rad
Angle ∠ B = β = 55.1550095421° = 55°9' = 0.96325507479 rad
Angle ∠ C = γ = 69.76998091581° = 69°41'59″ = 1.21664911578 rad

Height: ha = 19.69656433596
Height: hb = 19.69656433596
Height: hc = 17.23436879396

Median: ma = 19.95662020435
Median: mb = 19.95662020435
Median: mc = 17.23436879396

Inradius: r = 6.26767956144
Circumradius: R = 12.79547077127

Vertex coordinates: A[24; 0] B[0; 0] C[12; 17.23436879396]
Centroid: CG[12; 5.74545626465]
Coordinates of the circumscribed circle: U[12; 4.43989802269]
Coordinates of the inscribed circle: I[12; 6.26767956144]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.8549904579° = 124°51' = 0.96325507479 rad
∠ B' = β' = 124.8549904579° = 124°51' = 0.96325507479 rad
∠ C' = γ' = 110.3300190842° = 110°18'1″ = 1.21664911578 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 = 24 ; ;

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

p = a+b+c = 21+21+24 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-21)(33-21)(33-24) } ; ; T = sqrt{ 42768 } = 206.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 206.8 }{ 21 } = 19.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 206.8 }{ 21 } = 19.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 206.8 }{ 24 } = 17.23 ; ;

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{ 21**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 55° 9' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 55° 9' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 69° 41'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 206.8 }{ 33 } = 6.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 55° 9' } = 12.79 ; ;




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