Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 3.99993813446   b = 5.99990873357   c = 7.21

Area: T = 11.99663189873
Perimeter: p = 17.20884686802
Semiperimeter: s = 8.60442343401

Angle ∠ A = α = 33.69° = 33°41'24″ = 0.5888001425 rad
Angle ∠ B = β = 56.31° = 56°18'36″ = 0.98327949018 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 5.99990873357
Height: hb = 3.99993813446
Height: hc = 3.32876890395

Median: ma = 6.32435916729
Median: mb = 4.99992312763
Median: mc = 3.605

Inradius: r = 1.39442343401
Circumradius: R = 3.605

Vertex coordinates: A[7.21; 0] B[0; 0] C[2.21884536948; 3.32876890395]
Centroid: CG[3.14328178983; 1.10992296798]
Coordinates of the circumscribed circle: U[3.605; -0]
Coordinates of the inscribed circle: I[2.60551470045; 1.39442343401]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.31° = 146°18'36″ = 0.5888001425 rad
∠ B' = β' = 123.69° = 123°41'24″ = 0.98327949018 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 = 33° 41'24" ; ; beta = 56° 18'36" ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 33° 41'24" - 56° 18'36" = 90° ; ;

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

c = 7.21 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 7.21 * fraction{ sin 33° 41'24" }{ sin 90° } = 4 ; ;

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 = 7.21 * fraction{ sin 56° 18'36" }{ sin 90° } = 6 ; ;


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 = 4 ; ; b = 6 ; ; c = 7.21 ; ;

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

p = a+b+c = 4+6+7.21 = 17.21 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.21 }{ 2 } = 8.6 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.6 * (8.6-4)(8.6-6)(8.6-7.21) } ; ; T = sqrt{ 143.91 } = 12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12 }{ 4 } = 6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12 }{ 6 } = 4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12 }{ 7.21 } = 3.33 ; ;

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{ 6**2+7.21**2-4**2 }{ 2 * 6 * 7.21 } ) = 33° 41'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4**2+7.21**2-6**2 }{ 2 * 4 * 7.21 } ) = 56° 18'36" ; ; gamma = 180° - alpha - beta = 180° - 33° 41'24" - 56° 18'36" = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12 }{ 8.6 } = 1.39 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4 }{ 2 * sin 33° 41'24" } = 3.61 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6**2+2 * 7.21**2 - 4**2 } }{ 2 } = 6.324 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.21**2+2 * 4**2 - 6**2 } }{ 2 } = 4.999 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6**2+2 * 4**2 - 7.21**2 } }{ 2 } = 3.605 ; ;
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