18 20 21 triangle

Acute scalene triangle.

Sides: a = 18   b = 20   c = 21

Area: T = 165.5132650574
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 52.01334707109° = 52°48″ = 0.90878063193 rad
Angle ∠ B = β = 61.13112883899° = 61°7'53″ = 1.06769422584 rad
Angle ∠ C = γ = 66.85552408992° = 66°51'19″ = 1.16768440759 rad

Height: ha = 18.39902945082
Height: hb = 16.55112650574
Height: hc = 15.76331095785

Median: ma = 18.42655257727
Median: mb = 16.8087736314
Median: mc = 15.86766316526

Inradius: r = 5.61105983245
Circumradius: R = 11.41990667206

Vertex coordinates: A[21; 0] B[0; 0] C[8.69904761905; 15.76331095785]
Centroid: CG[9.89768253968; 5.25443698595]
Coordinates of the circumscribed circle: U[10.5; 4.48883276138]
Coordinates of the inscribed circle: I[9.5; 5.61105983245]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.9876529289° = 127°59'12″ = 0.90878063193 rad
∠ B' = β' = 118.869871161° = 118°52'7″ = 1.06769422584 rad
∠ C' = γ' = 113.1454759101° = 113°8'41″ = 1.16768440759 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 = 18 ; ; b = 20 ; ; c = 21 ; ;

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

p = a+b+c = 18+20+21 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-18)(29.5-20)(29.5-21) } ; ; T = sqrt{ 27394.44 } = 165.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 165.51 }{ 18 } = 18.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 165.51 }{ 20 } = 16.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 165.51 }{ 21 } = 15.76 ; ;

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{ 18**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 52° 48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 61° 7'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-18**2-20**2 }{ 2 * 20 * 18 } ) = 66° 51'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 165.51 }{ 29.5 } = 5.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 52° 48" } = 11.42 ; ;




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