Right triangle calculator (B,a,h)

Please enter two properties of the right triangle

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


You have entered cathetus a, height h and angle β.

Right scalene triangle.

Sides: a = 41   b = 67.2200031981   c = 78.72200374634

Area: T = 1377.601065561
Perimeter: p = 186.9220069444
Semiperimeter: s = 93.46600347222

Angle ∠ A = α = 31.38881498279° = 31°23'17″ = 0.54878265606 rad
Angle ∠ B = β = 58.61218501721° = 58°36'43″ = 1.02329697662 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 67.2200031981
Height: hb = 41
Height: hc = 35

Median: ma = 70.25773433759
Median: mb = 53.00990659658
Median: mc = 39.36600187317

Inradius: r = 14.74399972588
Circumradius: R = 39.36600187317

Vertex coordinates: A[78.72200374634; 0] B[0; 0] C[21.35441565041; 35]
Centroid: CG[33.35880646558; 11.66766666667]
Coordinates of the circumscribed circle: U[39.36600187317; 0]
Coordinates of the inscribed circle: I[26.26600027412; 14.74399972588]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.6121850172° = 148°36'43″ = 0.54878265606 rad
∠ B' = β' = 121.3888149828° = 121°23'17″ = 1.02329697662 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a, angle β and height h

a = 41 ; ; beta = 58.6° ; ; h = 35 ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 58.6 ° = 31.4 ° ; ;

3. From height h and cathetus a we calculate hypotenuse c - Euclid's theorem:

c_1**2 = a**2 - h**2 ; ; c_1 = sqrt{ 41**2 - 35**2 } = 21.354 ; ; ; ; c_1 c_2 = h**2 ; ; c_2 = h**2/c_1 = 35**2 / 21.354 = 57.366 ; ; ; ; c = c_1 + c_2 = 21.354 + 57.366 = 78.72 ; ;

4. 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{ 78.72**2 - 41**2 } = 67.2 ; ;


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 = 41 ; ; b = 67.2 ; ; c = 78.72 ; ;

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

p = a+b+c = 41+67.2+78.72 = 186.92 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 186.92 }{ 2 } = 93.46 ; ;

7. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 41 * 67.2 }{ 2 } = 1377.6 ; ;

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

h _a = b = 67.2 ; ; h _b = a = 41 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1377.6 }{ 78.72 } = 35 ; ;

9. 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{ 41 }{ 78.72 } ) = 31° 23'17" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 67.2 }{ 78.72 } ) = 58° 36'43" ; ; gamma = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1377.6 }{ 93.46 } = 14.74 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 41 }{ 2 * sin 31° 23'17" } = 39.36 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 67.2**2+2 * 78.72**2 - 41**2 } }{ 2 } = 70.257 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 78.72**2+2 * 41**2 - 67.2**2 } }{ 2 } = 53.009 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 67.2**2+2 * 41**2 - 78.72**2 } }{ 2 } = 39.36 ; ;
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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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