23 28 30 triangle

Acute scalene triangle.

Sides: a = 23   b = 28   c = 30

Area: T = 304.9977438514
Perimeter: p = 81
Semiperimeter: s = 40.5

Angle ∠ A = α = 46.56774634422° = 46°34'3″ = 0.81327555614 rad
Angle ∠ B = β = 62.13549067337° = 62°8'6″ = 1.08444587029 rad
Angle ∠ C = γ = 71.29876298241° = 71°17'51″ = 1.24443783893 rad

Height: ha = 26.52215163925
Height: hb = 21.78655313224
Height: hc = 20.33331625676

Median: ma = 26.64111336095
Median: mb = 22.77105950735
Median: mc = 20.77325780778

Inradius: r = 7.5310800951
Circumradius: R = 15.83661985712

Vertex coordinates: A[30; 0] B[0; 0] C[10.75; 20.33331625676]
Centroid: CG[13.58333333333; 6.77877208559]
Coordinates of the circumscribed circle: U[15; 5.07879114984]
Coordinates of the inscribed circle: I[12.5; 7.5310800951]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.4332536558° = 133°25'57″ = 0.81327555614 rad
∠ B' = β' = 117.8655093266° = 117°51'54″ = 1.08444587029 rad
∠ C' = γ' = 108.7022370176° = 108°42'9″ = 1.24443783893 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 = 30 ; ;

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

p = a+b+c = 23+28+30 = 81 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 81 }{ 2 } = 40.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.5 * (40.5-23)(40.5-28)(40.5-30) } ; ; T = sqrt{ 93023.44 } = 305 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 305 }{ 23 } = 26.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 305 }{ 28 } = 21.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 305 }{ 30 } = 20.33 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 28**2+30**2-23**2 }{ 2 * 28 * 30 } ) = 46° 34'3" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 23**2+30**2-28**2 }{ 2 * 23 * 30 } ) = 62° 8'6" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 23**2+28**2-30**2 }{ 2 * 23 * 28 } ) = 71° 17'51" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 305 }{ 40.5 } = 7.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 46° 34'3" } = 15.84 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 30**2 - 23**2 } }{ 2 } = 26.641 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 23**2 - 28**2 } }{ 2 } = 22.771 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 23**2 - 30**2 } }{ 2 } = 20.773 ; ;
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