23 30 30 triangle

Acute isosceles triangle.

Sides: a = 23   b = 30   c = 30

Area: T = 318.6455473057
Perimeter: p = 83
Semiperimeter: s = 41.5

Angle ∠ A = α = 45.0810620927° = 45°4'50″ = 0.7876805264 rad
Angle ∠ B = β = 67.46596895365° = 67°27'35″ = 1.17773936948 rad
Angle ∠ C = γ = 67.46596895365° = 67°27'35″ = 1.17773936948 rad

Height: ha = 27.7088302005
Height: hb = 21.24330315372
Height: hc = 21.24330315372

Median: ma = 27.7088302005
Median: mb = 22.12546468898
Median: mc = 22.12546468898

Inradius: r = 7.67882041701
Circumradius: R = 16.24106198662

Vertex coordinates: A[30; 0] B[0; 0] C[8.81766666667; 21.24330315372]
Centroid: CG[12.93988888889; 7.08110105124]
Coordinates of the circumscribed circle: U[15; 6.22655709487]
Coordinates of the inscribed circle: I[11.5; 7.67882041701]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.9199379073° = 134°55'10″ = 0.7876805264 rad
∠ B' = β' = 112.5440310464° = 112°32'25″ = 1.17773936948 rad
∠ C' = γ' = 112.5440310464° = 112°32'25″ = 1.17773936948 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 = 23 ; ; b = 30 ; ; c = 30 ; ;

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

p = a+b+c = 23+30+30 = 83 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 83 }{ 2 } = 41.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.5 * (41.5-23)(41.5-30)(41.5-30) } ; ; T = sqrt{ 101534.94 } = 318.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 318.65 }{ 23 } = 27.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 318.65 }{ 30 } = 21.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 318.65 }{ 30 } = 21.24 ; ;

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{ 23**2-30**2-30**2 }{ 2 * 30 * 30 } ) = 45° 4'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 30**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 67° 27'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-23**2-30**2 }{ 2 * 30 * 23 } ) = 67° 27'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 318.65 }{ 41.5 } = 7.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 45° 4'50" } = 16.24 ; ;




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