15 23 28 triangle

Obtuse scalene triangle.

Sides: a = 15   b = 23   c = 28

Area: T = 172.3376879396
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 32.35879982101° = 32°21'29″ = 0.56547536081 rad
Angle ∠ B = β = 55.1550095421° = 55°9' = 0.96325507479 rad
Angle ∠ C = γ = 92.4921906369° = 92°29'31″ = 1.61442882976 rad

Height: ha = 22.97882505862
Height: hb = 14.98658155997
Height: hc = 12.31097770997

Median: ma = 24.5
Median: mb = 19.29437813816
Median: mc = 13.45436240471

Inradius: r = 5.22223296787
Circumradius: R = 14.01332513044

Vertex coordinates: A[28; 0] B[0; 0] C[8.57114285714; 12.31097770997]
Centroid: CG[12.19904761905; 4.10332590332]
Coordinates of the circumscribed circle: U[14; -0.60992717958]
Coordinates of the inscribed circle: I[10; 5.22223296787]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.642200179° = 147°38'31″ = 0.56547536081 rad
∠ B' = β' = 124.8549904579° = 124°51' = 0.96325507479 rad
∠ C' = γ' = 87.5088093631° = 87°30'29″ = 1.61442882976 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 = 15 ; ; b = 23 ; ; c = 28 ; ;

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

p = a+b+c = 15+23+28 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-15)(33-23)(33-28) } ; ; T = sqrt{ 29700 } = 172.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 172.34 }{ 15 } = 22.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 172.34 }{ 23 } = 14.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 172.34 }{ 28 } = 12.31 ; ;

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+28**2-15**2 }{ 2 * 23 * 28 } ) = 32° 21'29" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15**2+28**2-23**2 }{ 2 * 15 * 28 } ) = 55° 9' ; ; gamma = 180° - alpha - beta = 180° - 32° 21'29" - 55° 9' = 92° 29'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 172.34 }{ 33 } = 5.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15 }{ 2 * sin 32° 21'29" } = 14.01 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 28**2 - 15**2 } }{ 2 } = 24.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 15**2 - 23**2 } }{ 2 } = 19.294 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 15**2 - 28**2 } }{ 2 } = 13.454 ; ;
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