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 cathetus a and perimeter p.

Right scalene triangle.

Sides: a = 84   b = 18.07769230769   c = 85.92330769231

Area: T = 759.2310769231
Perimeter: p = 188
Semiperimeter: s = 94

Angle ∠ A = α = 77.85550871856° = 77°51'18″ = 1.35988276108 rad
Angle ∠ B = β = 12.14549128144° = 12°8'42″ = 0.2121968716 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 18.07769230769
Height: hb = 84
Height: hc = 17.67223366159

Median: ma = 45.72549947833
Median: mb = 84.48548731252
Median: mc = 42.96215384615

Inradius: r = 8.07769230769
Circumradius: R = 42.96215384615

Vertex coordinates: A[85.92330769231; 0] B[0; 0] C[82.12199641898; 17.67223366159]
Centroid: CG[56.01443470376; 5.8910778872]
Coordinates of the circumscribed circle: U[42.96215384615; -0]
Coordinates of the inscribed circle: I[75.92330769231; 8.07769230769]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102.1454912814° = 102°8'42″ = 1.35988276108 rad
∠ B' = β' = 167.8555087186° = 167°51'18″ = 0.2121968716 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a and perimeter p

a = 84 ; ; p = 188 ; ;

2. From cathetus a and perimeter p we calculate cathetus b:

k_1 = p - a = b + c = 188-84 = 104 ; ; b = fraction{ k_1**2 - a**2 }{ 2 * k_1 } ; ; b = fraction{ 104**2 - 7056 }{ 2 * 104 } = 18.077 ; ;

3. From cathetus a we calculate hypotenuse c - Pythagorean theorem:

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 84**2 + 18.077**2 } = 85.923 ; ;
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 = 84 ; ; b = 18.08 ; ; c = 85.92 ; ;

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

p = a+b+c = 84+18.08+85.92 = 188 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 188 }{ 2 } = 94 ; ;

6. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 84 * 18.08 }{ 2 } = 759.23 ; ;

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

h _a = b = 18.08 ; ; h _b = a = 84 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 759.23 }{ 85.92 } = 17.67 ; ;

8. 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{ 84 }{ 85.92 } ) = 77° 51'18" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 18.08 }{ 85.92 } ) = 12° 8'42" ; ; gamma = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 759.23 }{ 94 } = 8.08 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 84 }{ 2 * sin 77° 51'18" } = 42.96 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.08**2+2 * 85.92**2 - 84**2 } }{ 2 } = 45.725 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 85.92**2+2 * 84**2 - 18.08**2 } }{ 2 } = 84.485 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.08**2+2 * 84**2 - 85.92**2 } }{ 2 } = 42.962 ; ;
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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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