9 30 30 triangle

Acute isosceles triangle.

Sides: a = 9   b = 30   c = 30

Area: T = 133.473260955
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 17.25438531174° = 17°15'14″ = 0.30111365456 rad
Angle ∠ B = β = 81.37330734413° = 81°22'23″ = 1.4220228054 rad
Angle ∠ C = γ = 81.37330734413° = 81°22'23″ = 1.4220228054 rad

Height: ha = 29.66105798999
Height: hb = 8.898817397
Height: hc = 8.898817397

Median: ma = 29.66105798999
Median: mb = 16.29441707368
Median: mc = 16.29441707368

Inradius: r = 3.86987712913
Circumradius: R = 15.17216521227

Vertex coordinates: A[30; 0] B[0; 0] C[1.35; 8.898817397]
Centroid: CG[10.45; 2.966605799]
Coordinates of the circumscribed circle: U[15; 2.27657478184]
Coordinates of the inscribed circle: I[4.5; 3.86987712913]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.7466146883° = 162°44'46″ = 0.30111365456 rad
∠ B' = β' = 98.62769265587° = 98°37'37″ = 1.4220228054 rad
∠ C' = γ' = 98.62769265587° = 98°37'37″ = 1.4220228054 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 = 30 ; ; c = 30 ; ;

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

p = a+b+c = 9+30+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 * (34.5-9)(34.5-30)(34.5-30) } ; ; T = sqrt{ 17814.94 } = 133.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 133.47 }{ 9 } = 29.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 133.47 }{ 30 } = 8.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 133.47 }{ 30 } = 8.9 ; ;

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**2-30**2-30**2 }{ 2 * 30 * 30 } ) = 17° 15'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 30**2-9**2-30**2 }{ 2 * 9 * 30 } ) = 81° 22'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-9**2-30**2 }{ 2 * 30 * 9 } ) = 81° 22'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 133.47 }{ 34.5 } = 3.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 17° 15'14" } = 15.17 ; ;




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