Right triangle calculator

Please enter two properties of the right triangle

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


You have entered cathetus b and angle α.

Right scalene triangle.

Sides: a = 4.12112161292   b = 1.5   c = 4.38657066002

Area: T = 3.09109120969
Perimeter: p = 10.00769227294
Semiperimeter: s = 5.00334613647

Angle ∠ A = α = 70° = 1.22217304764 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.5
Height: hb = 4.12112161292
Height: hc = 1.41095389312

Median: ma = 2.54987458869
Median: mb = 4.18989046759
Median: mc = 2.19328533001

Inradius: r = 0.61877547645
Circumradius: R = 2.19328533001

Vertex coordinates: A[4.38657066002; 0] B[0; 0] C[3.87326763853; 1.41095389312]
Centroid: CG[2.75327943285; 0.47698463104]
Coordinates of the circumscribed circle: U[2.19328533001; 0]
Coordinates of the inscribed circle: I[3.50334613647; 0.61877547645]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 110° = 1.22217304764 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b angle α

b = 1.5 ; ; alpha = 70° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 70 ° = 20 ° ; ;

3. From cathetus b and angle α we calculate hypotenuse c:

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(70 ° ) = 4.386 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(70 ° ) = 4.121 ; ;


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 = 4.12 ; ; b = 1.5 ; ; c = 4.39 ; ;

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

p = a+b+c = 4.12+1.5+4.39 = 10.01 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.01 }{ 2 } = 5 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5 * (5-4.12)(5-1.5)(5-4.39) } ; ; T = sqrt{ 9.55 } = 3.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.09 }{ 4.12 } = 1.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.09 }{ 1.5 } = 4.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.09 }{ 4.39 } = 1.41 ; ;

9. 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{ 4.12**2-1.5**2-4.39**2 }{ 2 * 1.5 * 4.39 } ) = 70° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.5**2-4.12**2-4.39**2 }{ 2 * 4.12 * 4.39 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.39**2-4.12**2-1.5**2 }{ 2 * 1.5 * 4.12 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.09 }{ 5 } = 0.62 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.12 }{ 2 * sin 70° } = 2.19 ; ;
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