Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, angle β and angle γ.

Acute scalene triangle.

Sides: a = 17.5   b = 17.5   c = 9.05986665786

Area: T = 76.56325
Perimeter: p = 44.05986665786
Semiperimeter: s = 22.02993332893

Angle ∠ A = α = 75° = 1.3098996939 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 30° = 0.52435987756 rad

Height: ha = 8.75
Height: hb = 8.75
Height: hc = 16.90437019601

Median: ma = 10.84439946556
Median: mb = 10.84439946556
Median: mc = 16.90437019601

Inradius: r = 3.47554796704
Circumradius: R = 9.05986665786

Vertex coordinates: A[9.05986665786; 0] B[0; 0] C[4.52993332893; 16.90437019601]
Centroid: CG[4.52993332893; 5.635456732]
Coordinates of the circumscribed circle: U[4.52993332893; 7.84550353815]
Coordinates of the inscribed circle: I[4.52993332893; 3.47554796704]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105° = 1.3098996939 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 150° = 0.52435987756 rad

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How did we calculate this triangle?

1. Input data entered: side a, angle β and angle γ.

a = 17.5 ; ; beta = 75° ; ; gamma = 30° ; ;

2. From angle β and angle γ we calculate angle α:

 beta + gamma + alpha = 180° ; ; alpha = 180° - beta - gamma = 180° - 75 ° - 30 ° = 75 ° ; ;

3. From angle β, angle α and side a we calculate b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 17.5 * fraction{ sin 75° }{ sin 75° } = 17.5 ; ;

4. From angle γ, angle α and side a we calculate c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 17.5 * fraction{ sin 30° }{ sin 75° } = 9.06 ; ;


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 = 17.5 ; ; b = 17.5 ; ; c = 9.06 ; ;

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

p = a+b+c = 17.5+17.5+9.06 = 44.06 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44.06 }{ 2 } = 22.03 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.03 * (22.03-17.5)(22.03-17.5)(22.03-9.06) } ; ; T = sqrt{ 5861.82 } = 76.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 76.56 }{ 17.5 } = 8.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 76.56 }{ 17.5 } = 8.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 76.56 }{ 9.06 } = 16.9 ; ;

9. 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{ 17.5**2+9.06**2-17.5**2 }{ 2 * 17.5 * 9.06 } ) = 75° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 17.5**2+9.06**2-17.5**2 }{ 2 * 17.5 * 9.06 } ) = 75° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 17.5**2+17.5**2-9.06**2 }{ 2 * 17.5 * 17.5 } ) = 30° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 76.56 }{ 22.03 } = 3.48 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17.5 }{ 2 * sin 75° } = 9.06 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.5**2+2 * 9.06**2 - 17.5**2 } }{ 2 } = 10.844 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.06**2+2 * 17.5**2 - 17.5**2 } }{ 2 } = 10.844 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.5**2+2 * 17.5**2 - 9.06**2 } }{ 2 } = 16.904 ; ;
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