9 27 27 triangle

Acute isosceles triangle.

Sides: a = 9   b = 27   c = 27

Area: T = 119.8010615608
Perimeter: p = 63
Semiperimeter: s = 31.5

Angle ∠ A = α = 19.18881364537° = 19°11'17″ = 0.33548961584 rad
Angle ∠ B = β = 80.40659317731° = 80°24'21″ = 1.40333482476 rad
Angle ∠ C = γ = 80.40659317731° = 80°24'21″ = 1.40333482476 rad

Height: ha = 26.62223590239
Height: hb = 8.87441196746
Height: hc = 8.87441196746

Median: ma = 26.62223590239
Median: mb = 14.92548115566
Median: mc = 14.92548115566

Inradius: r = 3.80331941463
Circumradius: R = 13.69114989266

Vertex coordinates: A[27; 0] B[0; 0] C[1.5; 8.87441196746]
Centroid: CG[9.5; 2.95880398915]
Coordinates of the circumscribed circle: U[13.5; 2.28219164878]
Coordinates of the inscribed circle: I[4.5; 3.80331941463]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.8121863546° = 160°48'43″ = 0.33548961584 rad
∠ B' = β' = 99.59440682269° = 99°35'39″ = 1.40333482476 rad
∠ C' = γ' = 99.59440682269° = 99°35'39″ = 1.40333482476 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 = 9 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 9+27+27 = 63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-9)(31.5-27)(31.5-27) } ; ; T = sqrt{ 14352.19 } = 119.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 119.8 }{ 9 } = 26.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 119.8 }{ 27 } = 8.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 119.8 }{ 27 } = 8.87 ; ;

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{ 9**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 19° 11'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-9**2-27**2 }{ 2 * 9 * 27 } ) = 80° 24'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-9**2-27**2 }{ 2 * 27 * 9 } ) = 80° 24'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 119.8 }{ 31.5 } = 3.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 19° 11'17" } = 13.69 ; ;




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