5 21 21 triangle

Acute isosceles triangle.

Sides: a = 5   b = 21   c = 21

Area: T = 52.1276648655
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 13.67442823363° = 13°40'27″ = 0.23986612496 rad
Angle ∠ B = β = 83.16328588319° = 83°9'46″ = 1.4511465702 rad
Angle ∠ C = γ = 83.16328588319° = 83°9'46″ = 1.4511465702 rad

Height: ha = 20.8510659462
Height: hb = 4.9644442729
Height: hc = 4.9644442729

Median: ma = 20.8510659462
Median: mb = 11.07992599031
Median: mc = 11.07992599031

Inradius: r = 2.21881552619
Circumradius: R = 10.57552050865

Vertex coordinates: A[21; 0] B[0; 0] C[0.59552380952; 4.9644442729]
Centroid: CG[7.19884126984; 1.6554814243]
Coordinates of the circumscribed circle: U[10.5; 1.25989529865]
Coordinates of the inscribed circle: I[2.5; 2.21881552619]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.3265717664° = 166°19'33″ = 0.23986612496 rad
∠ B' = β' = 96.83771411681° = 96°50'14″ = 1.4511465702 rad
∠ C' = γ' = 96.83771411681° = 96°50'14″ = 1.4511465702 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 = 5 ; ; b = 21 ; ; c = 21 ; ;

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

p = a+b+c = 5+21+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-5)(23.5-21)(23.5-21) } ; ; T = sqrt{ 2717.19 } = 52.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 52.13 }{ 5 } = 20.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 52.13 }{ 21 } = 4.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 52.13 }{ 21 } = 4.96 ; ;

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{ 5**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 13° 40'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-5**2-21**2 }{ 2 * 5 * 21 } ) = 83° 9'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-5**2-21**2 }{ 2 * 21 * 5 } ) = 83° 9'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 52.13 }{ 23.5 } = 2.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 13° 40'27" } = 10.58 ; ;




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