Triangle calculator ASA

Right isosceles triangle.

Sides: a = 22.6277416998 b = 22.6277416998 c = 32

Area: T = 256
Perimeter: p = 77.25548339959
Semiperimeter: s = 38.6277416998

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: h_a = 22.6277416998
Height: h_b = 22.6277416998
Height: h_c = 16

Median: m_a = 25.29882212813
Median: m_b = 25.29882212813
Median: m_c = 16

Inradius: r = 6.6277416998
Circumradius: R = 16

Vertex coordinates: A[32; 0] B[0; 0] C[16; 16]
Centroid: CG[16; 5.33333333333]
Coordinates of the circumscribed circle: U[16; -0]
Coordinates of the inscribed circle: I[16; 6.6277416998]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

$alpha = 45° ; ; beta = 45° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;$

2. By using the law of sines, we calculate unknown side a

$c = 32 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 32 * fraction{ sin 45° }{ sin 90° } = 22.63 ; ;$

3. By using the law of sines, we calculate last unknown side b

$fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 32 * fraction{ sin 45° }{ sin 90° } = 22.63 ; ;$
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.

$a = 22.63 ; ; b = 22.63 ; ; c = 32 ; ;$

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

$p = a+b+c = 22.63+22.63+32 = 77.25 ; ;$

5. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 77.25 }{ 2 } = 38.63 ; ;$

6. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.63 * (38.63-22.63)(38.63-22.63)(38.63-32) } ; ; T = sqrt{ 65536 } = 256 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 256 }{ 22.63 } = 22.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 256 }{ 22.63 } = 22.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 256 }{ 32 } = 16 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 22.63**2+32**2-22.63**2 }{ 2 * 22.63 * 32 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.63**2+32**2-22.63**2 }{ 2 * 22.63 * 32 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;$

9. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 256 }{ 38.63 } = 6.63 ; ;$

10. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 22.63 }{ 2 * sin 45° } = 16 ; ;$

11. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.63**2+2 * 32**2 - 22.63**2 } }{ 2 } = 25.298 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 22.63**2 - 22.63**2 } }{ 2 } = 25.298 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.63**2+2 * 22.63**2 - 32**2 } }{ 2 } = 16 ; ;$
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