27 28 29 triangle

Acute scalene triangle.

Sides: a = 27   b = 28   c = 29

Area: T = 338.6154825429
Perimeter: p = 84
Semiperimeter: s = 42

Angle ∠ A = α = 56.5154623377° = 56°30'53″ = 0.98663662535 rad
Angle ∠ B = β = 59.87331765842° = 59°52'23″ = 1.0454984065 rad
Angle ∠ C = γ = 63.61222000388° = 63°36'44″ = 1.11102423351 rad

Height: ha = 25.08325796614
Height: hb = 24.18767732449
Height: hc = 23.35327465813

Median: ma = 25.10547804213
Median: mb = 24.2699322199
Median: mc = 23.3721991785

Inradius: r = 8.06222577483
Circumradius: R = 16.18765328639

Vertex coordinates: A[29; 0] B[0; 0] C[13.55217241379; 23.35327465813]
Centroid: CG[14.1843908046; 7.78442488604]
Coordinates of the circumscribed circle: U[14.5; 7.19440146062]
Coordinates of the inscribed circle: I[14; 8.06222577483]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.4855376623° = 123°29'7″ = 0.98663662535 rad
∠ B' = β' = 120.1276823416° = 120°7'37″ = 1.0454984065 rad
∠ C' = γ' = 116.3887799961° = 116°23'16″ = 1.11102423351 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 = 27 ; ; b = 28 ; ; c = 29 ; ;

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

p = a+b+c = 27+28+29 = 84 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 84 }{ 2 } = 42 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42 * (42-27)(42-28)(42-29) } ; ; T = sqrt{ 114660 } = 338.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 338.61 }{ 27 } = 25.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 338.61 }{ 28 } = 24.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 338.61 }{ 29 } = 23.35 ; ;

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{ 27**2-28**2-29**2 }{ 2 * 28 * 29 } ) = 56° 30'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 59° 52'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-27**2-28**2 }{ 2 * 28 * 27 } ) = 63° 36'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 338.61 }{ 42 } = 8.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27 }{ 2 * sin 56° 30'53" } = 16.19 ; ;




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