10 20 27 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 20   c = 27

Area: T = 81.99904720074
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 17.6788070483° = 17°40'41″ = 0.30985405353 rad
Angle ∠ B = β = 37.39771865213° = 37°23'50″ = 0.65327040358 rad
Angle ∠ C = γ = 124.9254742996° = 124°55'29″ = 2.18803480825 rad

Height: ha = 16.39880944015
Height: hb = 8.19990472007
Height: hc = 6.07333682968

Median: ma = 23.22771392987
Median: mb = 17.7344147851
Median: mc = 8.23110388166

Inradius: r = 2.87768586669
Circumradius: R = 16.46553278234

Vertex coordinates: A[27; 0] B[0; 0] C[7.94444444444; 6.07333682968]
Centroid: CG[11.64881481481; 2.02444560989]
Coordinates of the circumscribed circle: U[13.5; -9.42664001789]
Coordinates of the inscribed circle: I[8.5; 2.87768586669]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.3221929517° = 162°19'19″ = 0.30985405353 rad
∠ B' = β' = 142.6032813479° = 142°36'10″ = 0.65327040358 rad
∠ C' = γ' = 55.07552570043° = 55°4'31″ = 2.18803480825 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 = 20 ; ; c = 27 ; ;

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

p = a+b+c = 10+20+27 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-10)(28.5-20)(28.5-27) } ; ; T = sqrt{ 6722.44 } = 81.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 81.99 }{ 10 } = 16.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 81.99 }{ 20 } = 8.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 81.99 }{ 27 } = 6.07 ; ;

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**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 17° 40'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-10**2-27**2 }{ 2 * 10 * 27 } ) = 37° 23'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-10**2-20**2 }{ 2 * 20 * 10 } ) = 124° 55'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 81.99 }{ 28.5 } = 2.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 17° 40'41" } = 16.47 ; ;




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