3 21 21 triangle

Acute isosceles triangle.

Sides: a = 3   b = 21   c = 21

Area: T = 31.42195400985
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 8.19220875163° = 8°11'32″ = 0.14329788998 rad
Angle ∠ B = β = 85.90439562418° = 85°54'14″ = 1.49993068769 rad
Angle ∠ C = γ = 85.90439562418° = 85°54'14″ = 1.49993068769 rad

Height: ha = 20.94663600657
Height: hb = 2.99223371522
Height: hc = 2.99223371522

Median: ma = 20.94663600657
Median: mb = 10.71221426428
Median: mc = 10.71221426428

Inradius: r = 1.39664240044
Circumradius: R = 10.52768886484

Vertex coordinates: A[21; 0] B[0; 0] C[0.21442857143; 2.99223371522]
Centroid: CG[7.07114285714; 0.99774457174]
Coordinates of the circumscribed circle: U[10.5; 0.75219206177]
Coordinates of the inscribed circle: I[1.5; 1.39664240044]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 171.8087912484° = 171°48'28″ = 0.14329788998 rad
∠ B' = β' = 94.09660437582° = 94°5'46″ = 1.49993068769 rad
∠ C' = γ' = 94.09660437582° = 94°5'46″ = 1.49993068769 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 = 3 ; ; b = 21 ; ; c = 21 ; ;

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

p = a+b+c = 3+21+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-3)(22.5-21)(22.5-21) } ; ; T = sqrt{ 987.19 } = 31.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.42 }{ 3 } = 20.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.42 }{ 21 } = 2.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.42 }{ 21 } = 2.99 ; ;

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{ 3**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 8° 11'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-3**2-21**2 }{ 2 * 3 * 21 } ) = 85° 54'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-3**2-21**2 }{ 2 * 21 * 3 } ) = 85° 54'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.42 }{ 22.5 } = 1.4 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 8° 11'32" } = 10.53 ; ;




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