Right triangle calculator (A)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered area T and angle α.

Right scalene triangle.

Sides: a = 55.0000297247   b = 76.9999583856   c = 8.60223086813

Area: T = 17.5
Perimeter: p = 20.60222967917
Semiperimeter: s = 10.30111483958

Angle ∠ A = α = 35.538° = 35°32'17″ = 0.62202551096 rad
Angle ∠ B = β = 54.462° = 54°27'43″ = 0.95105412172 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 76.9999583856
Height: hb = 55.0000297247
Height: hc = 4.06986752006

Median: ma = 7.43330001825
Median: mb = 6.10332902273
Median: mc = 4.30111543407

Inradius: r = 1.69988397145
Circumradius: R = 4.30111543407

Vertex coordinates: A[8.60223086813; 0] B[0; 0] C[2.90662311264; 4.06986752006]
Centroid: CG[3.83661799359; 1.35662250669]
Coordinates of the circumscribed circle: U[4.30111543407; 0]
Coordinates of the inscribed circle: I[3.30111900102; 1.69988397145]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.462° = 144°27'43″ = 0.62202551096 rad
∠ B' = β' = 125.538° = 125°32'17″ = 0.95105412172 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: angle α area T

 alpha = 35.538° ; ; S = 17.5 ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 35.538 ° = 54.462 ° ; ;

3. From area T and we calculate h:

S = fraction{ c * h }{ 2 } ; ; h = 2 * S / c = 2 * 17.5 / c = 4.069 ; ;

4. From and angle α we calculate cathetus a:

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(35.538 ° ) = 5 ; ;

5. From cathetus a and we calculate cathetus b - Pythagorean theorem:

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 8.602**2 - 5**2 } = 7 ; ;


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 = 5 ; ; b = 7 ; ; c = 8.6 ; ;

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

p = a+b+c = 5+7+8.6 = 20.6 ; ;

7. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.6 }{ 2 } = 10.3 ; ;

8. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.3 * (10.3-5)(10.3-7)(10.3-8.6) } ; ; T = sqrt{ 306.25 } = 17.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.5 }{ 5 } = 7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.5 }{ 7 } = 5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.5 }{ 8.6 } = 4.07 ; ;

10. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 5**2-7**2-8.6**2 }{ 2 * 7 * 8.6 } ) = 35° 32'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7**2-5**2-8.6**2 }{ 2 * 5 * 8.6 } ) = 54° 27'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.6**2-5**2-7**2 }{ 2 * 7 * 5 } ) = 90° ; ;

11. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.5 }{ 10.3 } = 1.7 ; ;

12. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 35° 32'17" } = 4.3 ; ;
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