Triangle calculator

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

You have entered side b, c and angle α.

Right scalene triangle.

Sides: a = 51467.90771364   b = 2584   c = 51403

Area: T = 66412676
Perimeter: p = 105454.9077136
Semiperimeter: s = 52727.45435682

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 2.87878042684° = 2°52'40″ = 0.05502271597 rad
Angle ∠ C = γ = 87.12221957316° = 87°7'20″ = 1.52105691671 rad

Height: ha = 2580.741126947
Height: hb = 51403
Height: hc = 2584

Median: ma = 25733.95435682
Median: mb = 51419.23444653
Median: mc = 25831.07696304

Inradius: r = 1259.546643181
Circumradius: R = 25733.95435682

Vertex coordinates: A[51403; 0] B[0; 0] C[51403; 2584]
Centroid: CG[34268.66766667; 861.3333333333]
Coordinates of the circumscribed circle: U[25701.5; 1292]
Coordinates of the inscribed circle: I[50143.45435682; 1259.546643181]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 177.1222195732° = 177°7'20″ = 0.05502271597 rad
∠ C' = γ' = 92.87878042684° = 92°52'40″ = 1.52105691671 rad

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How did we calculate this triangle?

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

b = 2584 ; ; c = 51403 ; ; alpha = 90° ; ;

2. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 2584**2+51403**2 - 2 * 2584 * 51403 * cos 90° } ; ; a = 51467.91 ; ;
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 = 51467.91 ; ; b = 2584 ; ; c = 51403 ; ;

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

p = a+b+c = 51467.91+2584+51403 = 105454.91 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 105454.91 }{ 2 } = 52727.45 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 52727.45 * (52727.45-51467.91)(52727.45-2584)(52727.45-51403) } ; ; T = sqrt{ 4.411 * 10**{ 15 } } = 66412676 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 66412676 }{ 51467.91 } = 2580.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 66412676 }{ 2584 } = 51403 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 66412676 }{ 51403 } = 2584 ; ;

7. 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{ 2584**2+51403**2-51467.91**2 }{ 2 * 2584 * 51403 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 51467.91**2+51403**2-2584**2 }{ 2 * 51467.91 * 51403 } ) = 2° 52'40" ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 2° 52'40" = 87° 7'20" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 66412676 }{ 52727.45 } = 1259.55 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 51467.91 }{ 2 * sin 90° } = 25733.95 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2584**2+2 * 51403**2 - 51467.91**2 } }{ 2 } = 25733.954 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 51403**2+2 * 51467.91**2 - 2584**2 } }{ 2 } = 51419.234 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2584**2+2 * 51467.91**2 - 51403**2 } }{ 2 } = 25831.07 ; ;
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