Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Right scalene triangle.

Sides: a = 69.06329490347   b = 84.10552372292   c = 48

Area: T = 1657.511077683
Perimeter: p = 201.1688186264
Semiperimeter: s = 100.5844093132

Angle ∠ A = α = 55.2° = 55°12' = 0.96334217471 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 34.8° = 34°48' = 0.60773745797 rad

Height: ha = 48
Height: hb = 39.41551620384
Height: hc = 69.06329490347

Median: ma = 59.1310556672
Median: mb = 42.05326186146
Median: mc = 73.11442320576

Inradius: r = 16.47988559028
Circumradius: R = 42.05326186146

Vertex coordinates: A[48; 0] B[0; 0] C[-0; 69.06329490347]
Centroid: CG[16; 23.02109830116]
Coordinates of the circumscribed circle: U[24; 34.53114745174]
Coordinates of the inscribed circle: I[16.47988559028; 16.47988559028]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.8° = 124°48' = 0.96334217471 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 145.2° = 145°12' = 0.60773745797 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 55° 12' ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 55° 12' - 90° = 34° 48' ; ;

2. By using the law of sines, we calculate unknown side a

c = 48 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 48 * fraction{ sin(55° 12') }{ sin (34° 48') } = 69.06 ; ;

3. By using the law of sines, we calculate last unknown side b

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 48 * fraction{ sin(90° ) }{ sin (34° 48') } = 84.11 ; ;


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 = 69.06 ; ; b = 84.11 ; ; c = 48 ; ;

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

p = a+b+c = 69.06+84.11+48 = 201.17 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 201.17 }{ 2 } = 100.58 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.58 * (100.58-69.06)(100.58-84.11)(100.58-48) } ; ; T = sqrt{ 2747341.98 } = 1657.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1657.51 }{ 69.06 } = 48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1657.51 }{ 84.11 } = 39.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1657.51 }{ 48 } = 69.06 ; ;

8. 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{ 69.06**2-84.11**2-48**2 }{ 2 * 84.11 * 48 } ) = 55° 12' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 84.11**2-69.06**2-48**2 }{ 2 * 69.06 * 48 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48**2-69.06**2-84.11**2 }{ 2 * 84.11 * 69.06 } ) = 34° 48' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1657.51 }{ 100.58 } = 16.48 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 69.06 }{ 2 * sin 55° 12' } = 42.05 ; ;




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