Right triangle calculator (A,S)

Please enter two properties of the right triangle

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


You have entered area S and angle α.

Right scalene triangle.

Sides: a = 5.09766953733   b = 7.84882226365   c = 9.35879325858

Area: T = 20
Perimeter: p = 22.30328505956
Semiperimeter: s = 11.15114252978

Angle ∠ A = α = 33° = 0.57659586532 rad
Angle ∠ B = β = 57° = 0.99548376736 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 7.84882226365
Height: hb = 5.09766953733
Height: hc = 4.27444484033

Median: ma = 8.25215861799
Median: mb = 6.43223365402
Median: mc = 4.67989662929

Inradius: r = 1.7933492712
Circumradius: R = 4.67989662929

Vertex coordinates: A[9.35879325858; 0] B[0; 0] C[2.77658592499; 4.27444484033]
Centroid: CG[4.04545972786; 1.42548161344]
Coordinates of the circumscribed circle: U[4.67989662929; 0]
Coordinates of the inscribed circle: I[3.30332026613; 1.7933492712]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147° = 0.57659586532 rad
∠ B' = β' = 123° = 0.99548376736 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: angle α and area S

 alpha = 33° ; ; S = 20 ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 33 ° = 57 ° ; ;

3. From area S, angle α and angle β we calculate hypotenuse c:

c**2 sin alpha sin beta = 2 S ; ; c = sqrt{ fraction{ 2 S }{ sin alpha sin beta } } ; ; c = sqrt{ fraction{ 2 * 20 }{ sin 33° * sin 57° } } = 9.358 ; ;

4. From area S and hypotenuse c we calculate height h:

S = fraction{ c * h }{ 2 } ; ; h = 2 * S / c = 2 * 20 / 9.358 = 4.274 ; ;

5. From hypotenuse c and angle α we calculate cathetus a:

 sin alpha = a:c ; ; a = c * sin alpha = 9.358 * sin(33 ° ) = 5.097 ; ;

6. From cathetus a and hypotenuse c we calculate cathetus b - Pythagorean theorem:

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 9.358**2 - 5.097**2 } = 7.848 ; ;
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 = 5.1 ; ; b = 7.85 ; ; c = 9.36 ; ;

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

p = a+b+c = 5.1+7.85+9.36 = 22.3 ; ;

8. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.3 }{ 2 } = 11.15 ; ;

9. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 5.1 * 7.85 }{ 2 } = 20 ; ;

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

h _a = b = 7.85 ; ; h _b = a = 5.1 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20 }{ 9.36 } = 4.27 ; ;

11. Calculation of the inner angles of the triangle - basic use of sine function

 sin alpha = fraction{ a }{ c } ; ; alpha = arcsin( fraction{ a }{ c } ) = arcsin( fraction{ 5.1 }{ 9.36 } ) = 33° ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 7.85 }{ 9.36 } ) = 57° ; ; gamma = 90° ; ;

12. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20 }{ 11.15 } = 1.79 ; ;

13. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.1 }{ 2 * sin 33° } = 4.68 ; ;

14. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.85**2+2 * 9.36**2 - 5.1**2 } }{ 2 } = 8.252 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.36**2+2 * 5.1**2 - 7.85**2 } }{ 2 } = 6.432 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.85**2+2 * 5.1**2 - 9.36**2 } }{ 2 } = 4.679 ; ;
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Trigonometry right triangle solver. You can calculate angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height. A right-angled triangle is entirely determined by two independent properties. Step-by-step explanations are provided for each calculation.

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