18 21 28 triangle

Obtuse scalene triangle.

Sides: a = 18   b = 21   c = 28

Area: T = 188.9440301418
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 39.99900139499° = 39°59'24″ = 0.69879574113 rad
Angle ∠ B = β = 48.57698611237° = 48°34'11″ = 0.84877039938 rad
Angle ∠ C = γ = 91.44401249264° = 91°26'24″ = 1.59659312484 rad

Height: ha = 20.99333668242
Height: hb = 17.99443144208
Height: hc = 13.49657358156

Median: ma = 23.05442837668
Median: mb = 21.06553744329
Median: mc = 13.65765002837

Inradius: r = 5.64400089976
Circumradius: R = 14.00444235144

Vertex coordinates: A[28; 0] B[0; 0] C[11.91107142857; 13.49657358156]
Centroid: CG[13.30435714286; 4.49985786052]
Coordinates of the circumscribed circle: U[14; -0.35219630248]
Coordinates of the inscribed circle: I[12.5; 5.64400089976]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.010998605° = 140°36″ = 0.69879574113 rad
∠ B' = β' = 131.4330138876° = 131°25'49″ = 0.84877039938 rad
∠ C' = γ' = 88.56598750736° = 88°33'36″ = 1.59659312484 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 = 21 ; ; c = 28 ; ;

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

p = a+b+c = 18+21+28 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-18)(33.5-21)(33.5-28) } ; ; T = sqrt{ 35698.44 } = 188.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 188.94 }{ 18 } = 20.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 188.94 }{ 21 } = 17.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 188.94 }{ 28 } = 13.5 ; ;

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**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 39° 59'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-18**2-28**2 }{ 2 * 18 * 28 } ) = 48° 34'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-18**2-21**2 }{ 2 * 21 * 18 } ) = 91° 26'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 188.94 }{ 33.5 } = 5.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 39° 59'24" } = 14 ; ;




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