Triangle calculator ASA

Acute scalene triangle.

Sides: a = 24.81097805026 b = 29.58553420209 c = 26.7

Area: T = 311.2366128283
Perimeter: p = 81.09551225235
Semiperimeter: s = 40.54875612618

Angle ∠ A = α = 52° = 0.9087571211 rad
Angle ∠ B = β = 70° = 1.22217304764 rad
Angle ∠ C = γ = 58° = 1.01222909662 rad

Height: h_a = 25.0989792975
Height: h_b = 21.04398871213
Height: h_c = 23.31435676616

Median: m_a = 25.30223700293
Median: m_b = 21.10441343976
Median: m_c = 23.81656741568

Inradius: r = 7.67658285479
Circumradius: R = 15.74220316849

Vertex coordinates: A[26.7; 0] B[0; 0] C[8.48554446834; 23.31435676616]
Centroid: CG[11.72884815611; 7.77111892205]
Coordinates of the circumscribed circle: U[13.35; 8.3422005848]
Coordinates of the inscribed circle: I[10.96222192408; 7.67658285479]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128° = 0.9087571211 rad
∠ B' = β' = 110° = 1.22217304764 rad
∠ C' = γ' = 122° = 1.01222909662 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

$alpha = 52° ; ; beta = 70° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 52° - 70° = 58° ; ;$

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

$c = 26.7 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 26.7 * fraction{ sin 52° }{ sin 58° } = 24.81 ; ;$

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

$fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 26.7 * fraction{ sin 70° }{ sin 58° } = 29.59 ; ;$

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.

$a = 24.81 ; ; b = 29.59 ; ; c = 26.7 ; ;$

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

$p = a+b+c = 24.81+29.59+26.7 = 81.1 ; ;$

5. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 81.1 }{ 2 } = 40.55 ; ;$

6. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.55 * (40.55-24.81)(40.55-29.59)(40.55-26.7) } ; ; T = sqrt{ 96867.93 } = 311.24 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 311.24 }{ 24.81 } = 25.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 311.24 }{ 29.59 } = 21.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 311.24 }{ 26.7 } = 23.31 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 29.59**2+26.7**2-24.81**2 }{ 2 * 29.59 * 26.7 } ) = 52° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24.81**2+26.7**2-29.59**2 }{ 2 * 24.81 * 26.7 } ) = 70° ; ; gamma = 180° - alpha - beta = 180° - 52° - 70° = 58° ; ;$

9. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 311.24 }{ 40.55 } = 7.68 ; ;$

10. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 24.81 }{ 2 * sin 52° } = 15.74 ; ;$

11. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.59**2+2 * 26.7**2 - 24.81**2 } }{ 2 } = 25.302 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 26.7**2+2 * 24.81**2 - 29.59**2 } }{ 2 } = 21.104 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.59**2+2 * 24.81**2 - 26.7**2 } }{ 2 } = 23.816 ; ;$
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