Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 0.6032885683   b = 2.32993714059   c = 2.25

Area: T = 0.67882463933
Perimeter: p = 5.18222570889
Semiperimeter: s = 2.59111285444

Angle ∠ A = α = 15° = 0.26217993878 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 75° = 1.3098996939 rad

Height: ha = 2.25
Height: hb = 0.58223428515
Height: hc = 0.6032885683

Median: ma = 2.27701030344
Median: mb = 1.1654685703
Median: mc = 1.27663605081

Inradius: r = 0.26217571385
Circumradius: R = 1.1654685703

Vertex coordinates: A[2.25; 0] B[0; 0] C[0; 0.6032885683]
Centroid: CG[0.75; 0.20109618943]
Coordinates of the circumscribed circle: U[1.125; 0.30114428415]
Coordinates of the inscribed circle: I[0.26217571385; 0.26217571385]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165° = 0.26217993878 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 105° = 1.3098996939 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 15° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 15° - 90° = 75° ; ;

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

c = 2.25 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 2.25 * fraction{ sin 15° }{ sin 75° } = 0.6 ; ;

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 = 2.25 * fraction{ sin 90° }{ sin 75° } = 2.33 ; ;


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 = 0.6 ; ; b = 2.33 ; ; c = 2.25 ; ;

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

p = a+b+c = 0.6+2.33+2.25 = 5.18 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.18 }{ 2 } = 2.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.59 * (2.59-0.6)(2.59-2.33)(2.59-2.25) } ; ; T = sqrt{ 0.46 } = 0.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.68 }{ 0.6 } = 2.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.68 }{ 2.33 } = 0.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.68 }{ 2.25 } = 0.6 ; ;

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{ 2.33**2+2.25**2-0.6**2 }{ 2 * 2.33 * 2.25 } ) = 15° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 0.6**2+2.25**2-2.33**2 }{ 2 * 0.6 * 2.25 } ) = 90° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 0.6**2+2.33**2-2.25**2 }{ 2 * 0.6 * 2.33 } ) = 75° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.68 }{ 2.59 } = 0.26 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 0.6 }{ 2 * sin 15° } = 1.16 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.33**2+2 * 2.25**2 - 0.6**2 } }{ 2 } = 2.27 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.25**2+2 * 0.6**2 - 2.33**2 } }{ 2 } = 1.165 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.33**2+2 * 0.6**2 - 2.25**2 } }{ 2 } = 1.276 ; ;
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