10 24 27 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 24   c = 27

Area: T = 119.2666246273
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 21.59988381421° = 21°35'56″ = 0.37769708402 rad
Angle ∠ B = β = 62.0621826732° = 62°3'43″ = 1.08331832163 rad
Angle ∠ C = γ = 96.33993351259° = 96°20'22″ = 1.68114385971 rad

Height: ha = 23.85332492546
Height: hb = 9.93988538561
Height: hc = 8.83545367609

Median: ma = 25.05499500998
Median: mb = 16.44768842034
Median: mc = 12.48799839743

Inradius: r = 3.91103687303
Circumradius: R = 13.58330551445

Vertex coordinates: A[27; 0] B[0; 0] C[4.68551851852; 8.83545367609]
Centroid: CG[10.56217283951; 2.9454845587]
Coordinates of the circumscribed circle: U[13.5; -1.54997956722]
Coordinates of the inscribed circle: I[6.5; 3.91103687303]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.4011161858° = 158°24'4″ = 0.37769708402 rad
∠ B' = β' = 117.9388173268° = 117°56'17″ = 1.08331832163 rad
∠ C' = γ' = 83.66106648741° = 83°39'38″ = 1.68114385971 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 = 10 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 10+24+27 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-10)(30.5-24)(30.5-27) } ; ; T = sqrt{ 14224.44 } = 119.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 119.27 }{ 10 } = 23.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 119.27 }{ 24 } = 9.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 119.27 }{ 27 } = 8.83 ; ;

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{ 10**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 21° 35'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-10**2-27**2 }{ 2 * 10 * 27 } ) = 62° 3'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-10**2-24**2 }{ 2 * 24 * 10 } ) = 96° 20'22" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 119.27 }{ 30.5 } = 3.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 21° 35'56" } = 13.58 ; ;




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