Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side c, angle α and angle β.

Right scalene triangle.

Sides: a = 1785.947713048   b = 683.4522377916   c = 1650

Area: T = 563848.212178
Perimeter: p = 4119.43995084
Semiperimeter: s = 2059.76997542

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 22.5° = 22°30' = 0.39326990817 rad
Angle ∠ C = γ = 67.5° = 67°30' = 1.17880972451 rad

Height: ha = 631.4287663402
Height: hb = 1650
Height: hc = 683.4522377916

Median: ma = 892.9743565241
Median: mb = 1685.015536735
Median: mc = 1071.323261849

Inradius: r = 273.7532623717
Circumradius: R = 892.9743565241

Vertex coordinates: A[1650; 0] B[0; 0] C[1650; 683.4522377916]
Centroid: CG[1100; 227.8177459305]
Coordinates of the circumscribed circle: U[825; 341.7266188958]
Coordinates of the inscribed circle: I[1376.247737628; 273.7532623717]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 157.5° = 157°30' = 0.39326990817 rad
∠ C' = γ' = 112.5° = 112°30' = 1.17880972451 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle α and angle β.

c = 1650 ; ; alpha = 90° ; ; beta = 22.5° ; ;

2. From angle α and angle β we calculate γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 22.5 ° = 67.5 ° ; ;

3. From angle α, angle γ and side c we calculate a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 1650 * fraction{ sin(90° ) }{ sin (67° 30') } = 1785.95 ; ;

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 1785.95**2+1650**2 - 2 * 1785.95 * 1650 * cos(22° 30') } ; ; b = 683.45 ; ;


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 = 1785.95 ; ; b = 683.45 ; ; c = 1650 ; ;

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

p = a+b+c = 1785.95+683.45+1650 = 4119.4 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4119.4 }{ 2 } = 2059.7 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2059.7 * (2059.7-1785.95)(2059.7-683.45)(2059.7-1650) } ; ; T = sqrt{ 317924805928 } = 563848.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 563848.21 }{ 1785.95 } = 631.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 563848.21 }{ 683.45 } = 1650 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 563848.21 }{ 1650 } = 683.45 ; ;

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{ 1785.95**2-683.45**2-1650**2 }{ 2 * 683.45 * 1650 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 683.45**2-1785.95**2-1650**2 }{ 2 * 1785.95 * 1650 } ) = 22° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1650**2-1785.95**2-683.45**2 }{ 2 * 683.45 * 1785.95 } ) = 67° 30' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 563848.21 }{ 2059.7 } = 273.75 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1785.95 }{ 2 * sin 90° } = 892.97 ; ;




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