Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Acute isosceles triangle.

Sides: a = 109.9099240141   b = 102   c = 102

Area: T = 4722.258844333
Perimeter: p = 313.9099240141
Semiperimeter: s = 156.9554620071

Angle ∠ A = α = 65.2° = 65°12' = 1.13879546723 rad
Angle ∠ B = β = 57.4° = 57°24' = 1.00218189906 rad
Angle ∠ C = γ = 57.4° = 57°24' = 1.00218189906 rad

Height: ha = 85.93301444947
Height: hb = 92.59333028104
Height: hc = 92.59333028104

Median: ma = 85.93301444947
Median: mb = 92.9577089747
Median: mc = 92.9577089747

Inradius: r = 30.08767756629
Circumradius: R = 60.53875451256

Vertex coordinates: A[102; 0] B[0; 0] C[59.21658875905; 92.59333028104]
Centroid: CG[53.73986291968; 30.86444342701]
Coordinates of the circumscribed circle: U[51; 32.61658607097]
Coordinates of the inscribed circle: I[54.95546200707; 30.08767756629]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114.8° = 114°48' = 1.13879546723 rad
∠ B' = β' = 122.6° = 122°36' = 1.00218189906 rad
∠ C' = γ' = 122.6° = 122°36' = 1.00218189906 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 65° 12' ; ; beta = 57° 24' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 65° 12' - 57° 24' = 57° 24' ; ;

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

c = 102 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 102 * fraction{ sin 65° 12' }{ sin 57° 24' } = 109.91 ; ;

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 = 102 * fraction{ sin 57° 24' }{ sin 57° 24' } = 102 ; ;


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 = 109.91 ; ; b = 102 ; ; c = 102 ; ;

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

p = a+b+c = 109.91+102+102 = 313.91 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 313.91 }{ 2 } = 156.95 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 156.95 * (156.95-109.91)(156.95-102)(156.95-102) } ; ; T = sqrt{ 22299724.81 } = 4722.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4722.26 }{ 109.91 } = 85.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4722.26 }{ 102 } = 92.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4722.26 }{ 102 } = 92.59 ; ;

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{ 102**2+102**2-109.91**2 }{ 2 * 102 * 102 } ) = 65° 12' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 109.91**2+102**2-102**2 }{ 2 * 109.91 * 102 } ) = 57° 24' ; ; gamma = 180° - alpha - beta = 180° - 65° 12' - 57° 24' = 57° 24' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4722.26 }{ 156.95 } = 30.09 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 109.91 }{ 2 * sin 65° 12' } = 60.54 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 102**2+2 * 102**2 - 109.91**2 } }{ 2 } = 85.93 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 102**2+2 * 109.91**2 - 102**2 } }{ 2 } = 92.957 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 102**2+2 * 109.91**2 - 102**2 } }{ 2 } = 92.957 ; ;
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