23 28 29 triangle

Acute scalene triangle.

Sides: a = 23   b = 28   c = 29

Area: T = 299.6599732977
Perimeter: p = 80
Semiperimeter: s = 40

Angle ∠ A = α = 47.555541258° = 47°33'19″ = 0.83299985267 rad
Angle ∠ B = β = 63.94220257221° = 63°56'31″ = 1.11659988792 rad
Angle ∠ C = γ = 68.5032561698° = 68°30'9″ = 1.19655952477 rad

Height: ha = 26.05221506937
Height: hb = 21.43999809269
Height: hc = 20.66220505502

Median: ma = 26.08216027115
Median: mb = 22.11333443875
Median: mc = 21.12546301743

Inradius: r = 7.49899933244
Circumradius: R = 15.58441260391

Vertex coordinates: A[29; 0] B[0; 0] C[10.10334482759; 20.66220505502]
Centroid: CG[13.03444827586; 6.88773501834]
Coordinates of the circumscribed circle: U[14.5; 5.71109530205]
Coordinates of the inscribed circle: I[12; 7.49899933244]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.445458742° = 132°26'41″ = 0.83299985267 rad
∠ B' = β' = 116.0587974278° = 116°3'29″ = 1.11659988792 rad
∠ C' = γ' = 111.4977438302° = 111°29'51″ = 1.19655952477 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 = 23 ; ; b = 28 ; ; c = 29 ; ;

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

p = a+b+c = 23+28+29 = 80 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 80 }{ 2 } = 40 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40 * (40-23)(40-28)(40-29) } ; ; T = sqrt{ 89760 } = 299.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 299.6 }{ 23 } = 26.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 299.6 }{ 28 } = 21.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 299.6 }{ 29 } = 20.66 ; ;

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{ 23**2-28**2-29**2 }{ 2 * 28 * 29 } ) = 47° 33'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-23**2-29**2 }{ 2 * 23 * 29 } ) = 63° 56'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-23**2-28**2 }{ 2 * 28 * 23 } ) = 68° 30'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 299.6 }{ 40 } = 7.49 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 47° 33'19" } = 15.58 ; ;




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