16 23 30 triangle

Obtuse scalene triangle.

Sides: a = 16   b = 23   c = 30

Area: T = 181.7440027237
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 31.78883306171° = 31°47'18″ = 0.5554811033 rad
Angle ∠ B = β = 49.22223700378° = 49°13'21″ = 0.85990924228 rad
Angle ∠ C = γ = 98.98992993452° = 98°59'21″ = 1.72876891978 rad

Height: ha = 22.71875034046
Height: hb = 15.80334806293
Height: hc = 12.11660018158

Median: ma = 25.50549014897
Median: mb = 21.11327923307
Median: mc = 12.94221791055

Inradius: r = 5.26878268764
Circumradius: R = 15.1876527932

Vertex coordinates: A[30; 0] B[0; 0] C[10.45; 12.11660018158]
Centroid: CG[13.48333333333; 4.03986672719]
Coordinates of the circumscribed circle: U[15; -2.37328949894]
Coordinates of the inscribed circle: I[11.5; 5.26878268764]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.2121669383° = 148°12'42″ = 0.5554811033 rad
∠ B' = β' = 130.7787629962° = 130°46'39″ = 0.85990924228 rad
∠ C' = γ' = 81.01107006548° = 81°39″ = 1.72876891978 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 = 16 ; ; b = 23 ; ; c = 30 ; ;

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

p = a+b+c = 16+23+30 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-16)(34.5-23)(34.5-30) } ; ; T = sqrt{ 33029.44 } = 181.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 181.74 }{ 16 } = 22.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 181.74 }{ 23 } = 15.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 181.74 }{ 30 } = 12.12 ; ;

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{ 23**2+30**2-16**2 }{ 2 * 23 * 30 } ) = 31° 47'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16**2+30**2-23**2 }{ 2 * 16 * 30 } ) = 49° 13'21" ; ; gamma = 180° - alpha - beta = 180° - 31° 47'18" - 49° 13'21" = 98° 59'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 181.74 }{ 34.5 } = 5.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16 }{ 2 * sin 31° 47'18" } = 15.19 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 30**2 - 16**2 } }{ 2 } = 25.505 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 16**2 - 23**2 } }{ 2 } = 21.113 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 16**2 - 30**2 } }{ 2 } = 12.942 ; ;
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