Triangle calculator ASA

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


Obtuse scalene triangle.

Sides: a = 7.91549893473   b = 5.26771301924   c = 4.7

Area: T = 11.95659941456
Perimeter: p = 17.88221195396
Semiperimeter: s = 8.94110597698

Angle ∠ A = α = 105° = 1.83325957146 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: ha = 3.02111017655
Height: hb = 4.54398513836
Height: hc = 5.08876570832

Median: ma = 3.04221318413
Median: mb = 5.95325509715
Median: mc = 6.29985997186

Inradius: r = 1.3377201009
Circumradius: R = 4.09770999697

Vertex coordinates: A[4.7; 0] B[0; 0] C[6.06332336068; 5.08876570832]
Centroid: CG[3.58877445356; 1.69658856944]
Coordinates of the circumscribed circle: U[2.35; 3.35661478158]
Coordinates of the inscribed circle: I[3.67439295775; 1.3377201009]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 75° = 1.83325957146 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 145° = 0.61108652382 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 105° ; ; beta = 40° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 105° - 40° = 35° ; ;

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

c = 4.7 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 4.7 * fraction{ sin 105° }{ sin 35° } = 7.91 ; ;

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 = 4.7 * fraction{ sin 40° }{ sin 35° } = 5.27 ; ;


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 = 7.91 ; ; b = 5.27 ; ; c = 4.7 ; ;

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

p = a+b+c = 7.91+5.27+4.7 = 17.88 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.88 }{ 2 } = 8.94 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.94 * (8.94-7.91)(8.94-5.27)(8.94-4.7) } ; ; T = sqrt{ 142.95 } = 11.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.96 }{ 7.91 } = 3.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.96 }{ 5.27 } = 4.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.96 }{ 4.7 } = 5.09 ; ;

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{ 5.27**2+4.7**2-7.91**2 }{ 2 * 5.27 * 4.7 } ) = 105° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.91**2+4.7**2-5.27**2 }{ 2 * 7.91 * 4.7 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 105° - 40° = 35° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.96 }{ 8.94 } = 1.34 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.91 }{ 2 * sin 105° } = 4.1 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.27**2+2 * 4.7**2 - 7.91**2 } }{ 2 } = 3.042 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.7**2+2 * 7.91**2 - 5.27**2 } }{ 2 } = 5.953 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.27**2+2 * 7.91**2 - 4.7**2 } }{ 2 } = 6.299 ; ;
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