5 20 21 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 20   c = 21

Area: T = 49.84397431775
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 13.72991326754° = 13°43'45″ = 0.24396185686 rad
Angle ∠ B = β = 71.68223015888° = 71°40'56″ = 1.25110921781 rad
Angle ∠ C = γ = 94.58985657358° = 94°35'19″ = 1.65108819068 rad

Height: ha = 19.9365897271
Height: hb = 4.98439743178
Height: hc = 4.74766422074

Median: ma = 20.3533132437
Median: mb = 11.53325625947
Median: mc = 10.11218742081

Inradius: r = 2.16769453555
Circumradius: R = 10.5343762145

Vertex coordinates: A[21; 0] B[0; 0] C[1.57114285714; 4.74766422074]
Centroid: CG[7.52438095238; 1.58222140691]
Coordinates of the circumscribed circle: U[10.5; -0.84327009716]
Coordinates of the inscribed circle: I[3; 2.16769453555]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.2710867325° = 166°16'15″ = 0.24396185686 rad
∠ B' = β' = 108.3187698411° = 108°19'4″ = 1.25110921781 rad
∠ C' = γ' = 85.41114342642° = 85°24'41″ = 1.65108819068 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 5+20+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 * (23-5)(23-20)(23-21) } ; ; T = sqrt{ 2484 } = 49.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 49.84 }{ 5 } = 19.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 49.84 }{ 20 } = 4.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 49.84 }{ 21 } = 4.75 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 13° 43'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-5**2-21**2 }{ 2 * 5 * 21 } ) = 71° 40'56" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-5**2-20**2 }{ 2 * 20 * 5 } ) = 94° 35'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 49.84 }{ 23 } = 2.17 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 13° 43'45" } = 10.53 ; ;




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