Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Acute scalene triangle.

Sides: a = 3473   b = 4822   c = 5009.119890109

Area: T = 7973902.282167
Perimeter: p = 13304.11989011
Semiperimeter: s = 6652.059945054

Angle ∠ A = α = 41.31993999231° = 41°19'10″ = 0.72111595736 rad
Angle ∠ B = β = 66.45106000769° = 66°27'2″ = 1.16597817613 rad
Angle ∠ C = γ = 72.23° = 72°13'48″ = 1.26106513187 rad

Height: ha = 4591.939911988
Height: hb = 3307.301082193
Height: hc = 3183.754444429

Median: ma = 4599.577017912
Median: mb = 3572.615523014
Median: mc = 3374.002184628

Inradius: r = 1198.712181864
Circumradius: R = 2630.0440459

Vertex coordinates: A[5009.119890109; 0] B[0; 0] C[1387.60110372; 3183.754444429]
Centroid: CG[2132.243997943; 1061.251148143]
Coordinates of the circumscribed circle: U[2504.559945054; 802.6879746006]
Coordinates of the inscribed circle: I[1830.059945054; 1198.712181864]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.6810600077° = 138°40'50″ = 0.72111595736 rad
∠ B' = β' = 113.5499399923° = 113°32'58″ = 1.16597817613 rad
∠ C' = γ' = 107.77° = 107°46'12″ = 1.26106513187 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 3473 ; ; b = 4822 ; ; gamma = 72° 13'48" ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 3473**2+4822**2 - 2 * 3473 * 4822 * cos(72° 13'48") } ; ; c = 5009.12 ; ;


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 = 3473 ; ; b = 4822 ; ; c = 5009.12 ; ;

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

p = a+b+c = 3473+4822+5009.12 = 13304.12 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13304.12 }{ 2 } = 6652.06 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6652.06 * (6652.06-3473)(6652.06-4822)(6652.06-5009.12) } ; ; T = sqrt{ 6.358 * 10**{ 13 } } = 7973902.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7973902.28 }{ 3473 } = 4591.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7973902.28 }{ 4822 } = 3307.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7973902.28 }{ 5009.12 } = 3183.75 ; ;

6. 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{ 3473**2-4822**2-5009.12**2 }{ 2 * 4822 * 5009.12 } ) = 41° 19'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4822**2-3473**2-5009.12**2 }{ 2 * 3473 * 5009.12 } ) = 66° 27'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5009.12**2-3473**2-4822**2 }{ 2 * 4822 * 3473 } ) = 72° 13'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7973902.28 }{ 6652.06 } = 1198.71 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3473 }{ 2 * sin 41° 19'10" } = 2630.04 ; ;




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