12 21 21 triangle

Acute isosceles triangle.

Sides: a = 12   b = 21   c = 21

Area: T = 120.7487670785
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 33.2033099198° = 33°12'11″ = 0.58795034029 rad
Angle ∠ B = β = 73.3988450401° = 73°23'54″ = 1.28110446254 rad
Angle ∠ C = γ = 73.3988450401° = 73°23'54″ = 1.28110446254 rad

Height: ha = 20.12546117975
Height: hb = 11.549977817
Height: hc = 11.549977817

Median: ma = 20.12546117975
Median: mb = 13.5
Median: mc = 13.5

Inradius: r = 4.4722135955
Circumradius: R = 10.95767330897

Vertex coordinates: A[21; 0] B[0; 0] C[3.42985714286; 11.549977817]
Centroid: CG[8.14328571429; 3.833325939]
Coordinates of the circumscribed circle: U[10.5; 3.13304951685]
Coordinates of the inscribed circle: I[6; 4.4722135955]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.7976900802° = 146°47'49″ = 0.58795034029 rad
∠ B' = β' = 106.6021549599° = 106°36'6″ = 1.28110446254 rad
∠ C' = γ' = 106.6021549599° = 106°36'6″ = 1.28110446254 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 = 12 ; ; b = 21 ; ; c = 21 ; ;

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

p = a+b+c = 12+21+21 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-12)(27-21)(27-21) } ; ; T = sqrt{ 14580 } = 120.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 120.75 }{ 12 } = 20.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 120.75 }{ 21 } = 11.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 120.75 }{ 21 } = 11.5 ; ;

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{ 12**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 33° 12'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 73° 23'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-12**2-21**2 }{ 2 * 21 * 12 } ) = 73° 23'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 120.75 }{ 27 } = 4.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 33° 12'11" } = 10.96 ; ;




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