4 21 21 triangle

Acute isosceles triangle.

Sides: a = 4   b = 21   c = 21

Area: T = 41.80990899207
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 10.93300475998° = 10°55'48″ = 0.1910765318 rad
Angle ∠ B = β = 84.53549762001° = 84°32'6″ = 1.47554136678 rad
Angle ∠ C = γ = 84.53549762001° = 84°32'6″ = 1.47554136678 rad

Height: ha = 20.90545449604
Height: hb = 3.98218180877
Height: hc = 3.98218180877

Median: ma = 20.90545449604
Median: mb = 10.87442815855
Median: mc = 10.87442815855

Inradius: r = 1.81877865183
Circumradius: R = 10.54879454548

Vertex coordinates: A[21; 0] B[0; 0] C[0.3810952381; 3.98218180877]
Centroid: CG[7.1276984127; 1.32772726959]
Coordinates of the circumscribed circle: U[10.5; 1.00545662338]
Coordinates of the inscribed circle: I[2; 1.81877865183]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.07699524° = 169°4'12″ = 0.1910765318 rad
∠ B' = β' = 95.46550237999° = 95°27'54″ = 1.47554136678 rad
∠ C' = γ' = 95.46550237999° = 95°27'54″ = 1.47554136678 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 = 4 ; ; b = 21 ; ; c = 21 ; ;

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

p = a+b+c = 4+21+21 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-4)(23-21)(23-21) } ; ; T = sqrt{ 1748 } = 41.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.81 }{ 4 } = 20.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.81 }{ 21 } = 3.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.81 }{ 21 } = 3.98 ; ;

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{ 4**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 10° 55'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-4**2-21**2 }{ 2 * 4 * 21 } ) = 84° 32'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-4**2-21**2 }{ 2 * 21 * 4 } ) = 84° 32'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.81 }{ 23 } = 1.82 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 10° 55'48" } = 10.55 ; ;




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