23 23 30 triangle

Acute isosceles triangle.

Sides: a = 23   b = 23   c = 30

Area: T = 261.5343936612
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 49.2944293169° = 49°17'39″ = 0.86603477182 rad
Angle ∠ B = β = 49.2944293169° = 49°17'39″ = 0.86603477182 rad
Angle ∠ C = γ = 81.41114136621° = 81°24'41″ = 1.42108972171 rad

Height: ha = 22.74220814446
Height: hb = 22.74220814446
Height: hc = 17.43655957742

Median: ma = 24.1329857024
Median: mb = 24.1329857024
Median: mc = 17.43655957742

Inradius: r = 6.88224720161
Circumradius: R = 15.17701154022

Vertex coordinates: A[30; 0] B[0; 0] C[15; 17.43655957742]
Centroid: CG[15; 5.81218652581]
Coordinates of the circumscribed circle: U[15; 2.2655480372]
Coordinates of the inscribed circle: I[15; 6.88224720161]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.7065706831° = 130°42'21″ = 0.86603477182 rad
∠ B' = β' = 130.7065706831° = 130°42'21″ = 0.86603477182 rad
∠ C' = γ' = 98.58985863379° = 98°35'19″ = 1.42108972171 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 = 23 ; ; c = 30 ; ;

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

p = a+b+c = 23+23+30 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-23)(38-23)(38-30) } ; ; T = sqrt{ 68400 } = 261.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 261.53 }{ 23 } = 22.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 261.53 }{ 23 } = 22.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 261.53 }{ 30 } = 17.44 ; ;

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**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 49° 17'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 49° 17'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 81° 24'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 261.53 }{ 38 } = 6.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 49° 17'39" } = 15.17 ; ;




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