21 26 30 triangle

Acute scalene triangle.

Sides: a = 21   b = 26   c = 30

Area: T = 267.55554849
Perimeter: p = 77
Semiperimeter: s = 38.5

Angle ∠ A = α = 43.31774289986° = 43°19'3″ = 0.75660317595 rad
Angle ∠ B = β = 58.1454569176° = 58°8'40″ = 1.01548141743 rad
Angle ∠ C = γ = 78.53880018254° = 78°32'17″ = 1.37107467198 rad

Height: ha = 25.48114747524
Height: hb = 20.58111911461
Height: hc = 17.83770323267

Median: ma = 26.03436320939
Median: mb = 22.39441956766
Median: mc = 18.26219823677

Inradius: r = 6.94994931143
Circumradius: R = 15.30552365999

Vertex coordinates: A[30; 0] B[0; 0] C[11.08333333333; 17.83770323267]
Centroid: CG[13.69444444444; 5.94656774422]
Coordinates of the circumscribed circle: U[15; 3.04114252218]
Coordinates of the inscribed circle: I[12.5; 6.94994931143]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.6832571001° = 136°40'57″ = 0.75660317595 rad
∠ B' = β' = 121.8555430824° = 121°51'20″ = 1.01548141743 rad
∠ C' = γ' = 101.4621998175° = 101°27'43″ = 1.37107467198 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 = 21 ; ; b = 26 ; ; c = 30 ; ;

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

p = a+b+c = 21+26+30 = 77 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 77 }{ 2 } = 38.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.5 * (38.5-21)(38.5-26)(38.5-30) } ; ; T = sqrt{ 71585.94 } = 267.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 267.56 }{ 21 } = 25.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 267.56 }{ 26 } = 20.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 267.56 }{ 30 } = 17.84 ; ;

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{ 21**2-26**2-30**2 }{ 2 * 26 * 30 } ) = 43° 19'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 58° 8'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-21**2-26**2 }{ 2 * 26 * 21 } ) = 78° 32'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 267.56 }{ 38.5 } = 6.95 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 43° 19'3" } = 15.31 ; ;




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