Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 909.9022128364   b = 243.8087540487   c = 942

Area: T = 110920.5
Perimeter: p = 2095.710966885
Semiperimeter: s = 1047.855483443

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

Height: ha = 243.8087540487
Height: hb = 909.9022128364
Height: hc = 235.5

Median: ma = 516.161139685
Median: mb = 918.0321814482
Median: mc = 471

Inradius: r = 105.8554834425
Circumradius: R = 471

Vertex coordinates: A[942; 0] B[0; 0] C[878.8987965183; 235.5]
Centroid: CG[606.9665988394; 78.5]
Coordinates of the circumscribed circle: U[471; 0]
Coordinates of the inscribed circle: I[804.0477293939; 105.8554834425]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105° = 1.3098996939 rad
∠ B' = β' = 165° = 0.26217993878 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 = 75° ; ; beta = 15° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 75° - 15° = 90° ; ;

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

c = 942 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 942 * fraction{ sin 75° }{ sin 90° } = 909.9 ; ;

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 = 942 * fraction{ sin 15° }{ sin 90° } = 243.81 ; ;
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 = 909.9 ; ; b = 243.81 ; ; c = 942 ; ;

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

p = a+b+c = 909.9+243.81+942 = 2095.71 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2095.71 }{ 2 } = 1047.85 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1047.85 * (1047.85-909.9)(1047.85-243.81)(1047.85-942) } ; ; T = sqrt{ 12303357320.2 } = 110920.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 110920.5 }{ 909.9 } = 243.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 110920.5 }{ 243.81 } = 909.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 110920.5 }{ 942 } = 235.5 ; ;

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{ 243.81**2+942**2-909.9**2 }{ 2 * 243.81 * 942 } ) = 75° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 909.9**2+942**2-243.81**2 }{ 2 * 909.9 * 942 } ) = 15° ; ;
 gamma = 180° - alpha - beta = 180° - 75° - 15° = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 110920.5 }{ 1047.85 } = 105.85 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 909.9 }{ 2 * sin 75° } = 471 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 243.81**2+2 * 942**2 - 909.9**2 } }{ 2 } = 516.161 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 942**2+2 * 909.9**2 - 243.81**2 } }{ 2 } = 918.032 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 243.81**2+2 * 909.9**2 - 942**2 } }{ 2 } = 471 ; ;
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