Triangle calculator

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

You have entered side c, angle α and angle β.

Right scalene triangle.

Sides: a = 40.15503505446   b = 57.34106431002   c = 70

Area: T = 1151.123346046
Perimeter: p = 167.4910993645
Semiperimeter: s = 83.74554968224

Angle ∠ A = α = 35° = 0.61108652382 rad
Angle ∠ B = β = 55° = 0.96599310886 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 57.34106431002
Height: hb = 40.15503505446
Height: hc = 32.88992417275

Median: ma = 60.75332880868
Median: mb = 49.33659705148
Median: mc = 35

Inradius: r = 13.74554968224
Circumradius: R = 35

Vertex coordinates: A[70; 0] B[0; 0] C[23.02992949836; 32.88992417275]
Centroid: CG[31.01097649945; 10.96330805758]
Coordinates of the circumscribed circle: U[35; 0]
Coordinates of the inscribed circle: I[26.40548537222; 13.74554968224]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145° = 0.61108652382 rad
∠ B' = β' = 125° = 0.96599310886 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle α and angle β.

c = 70 ; ; alpha = 35° ; ; beta = 55° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 35 ° - 55 ° = 90 ° ; ;

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

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 70 * fraction{ sin 35° }{ sin 90° } = 40.15 ; ;

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 40.15**2+70**2 - 2 * 40.15 * 70 * cos 55° } ; ; b = 57.34 ; ;


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 = 40.15 ; ; b = 57.34 ; ; c = 70 ; ;

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

p = a+b+c = 40.15+57.34+70 = 167.49 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 167.49 }{ 2 } = 83.75 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 83.75 * (83.75-40.15)(83.75-57.34)(83.75-70) } ; ; T = sqrt{ 1325085.22 } = 1151.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1151.12 }{ 40.15 } = 57.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1151.12 }{ 57.34 } = 40.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1151.12 }{ 70 } = 32.89 ; ;

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{ 57.34**2+70**2-40.15**2 }{ 2 * 57.34 * 70 } ) = 35° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40.15**2+70**2-57.34**2 }{ 2 * 40.15 * 70 } ) = 55° ; ; gamma = 180° - alpha - beta = 180° - 35° - 55° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1151.12 }{ 83.75 } = 13.75 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40.15 }{ 2 * sin 35° } = 35 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 57.34**2+2 * 70**2 - 40.15**2 } }{ 2 } = 60.753 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 70**2+2 * 40.15**2 - 57.34**2 } }{ 2 } = 49.336 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 57.34**2+2 * 40.15**2 - 70**2 } }{ 2 } = 35 ; ;
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