27 27 30 triangle

Acute isosceles triangle.

Sides: a = 27 b = 27 c = 30

Area: T = 336.749916481
Perimeter: p = 84
Semiperimeter: s = 42

Angle ∠ A = α = 56.25110114041° = 56°15'4″ = 0.98217653566 rad
Angle ∠ B = β = 56.25110114041° = 56°15'4″ = 0.98217653566 rad
Angle ∠ C = γ = 67.49879771918° = 67°29'53″ = 1.17880619404 rad

Height: h_a = 24.94443825785
Height: h_b = 24.94443825785
Height: h_c = 22.45499443206

Median: m_a = 25.14545819214
Median: m_b = 25.14545819214
Median: m_c = 22.45499443206

Inradius: r = 8.01878372574
Circumradius: R = 16.23661204462

Vertex coordinates: A[30; 0] B[0; 0] C[15; 22.45499443206]
Centroid: CG[15; 7.48333147735]
Coordinates of the circumscribed circle: U[15; 6.21438238745]
Coordinates of the inscribed circle: I[15; 8.01878372574]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.7498988596° = 123°44'56″ = 0.98217653566 rad
∠ B' = β' = 123.7498988596° = 123°44'56″ = 0.98217653566 rad
∠ C' = γ' = 112.5022022808° = 112°30'7″ = 1.17880619404 rad

Calculate another triangle

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.

$a = 27 ; ; b = 27 ; ; c = 30 ; ;$

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

$p = a+b+c = 27+27+30 = 84 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 84 }{ 2 } = 42 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42 * (42-27)(42-27)(42-30) } ; ; T = sqrt{ 113400 } = 336.75 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 336.75 }{ 27 } = 24.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 336.75 }{ 27 } = 24.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 336.75 }{ 30 } = 22.45 ; ;$

5. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 27**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 56° 15'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 56° 15'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 67° 29'53" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 336.75 }{ 42 } = 8.02 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27 }{ 2 * sin 56° 15'4" } = 16.24 ; ;$

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