Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=35; b=9.69771987616; c=42 and a=35; b=55.58330620008; c=42.

#1 Obtuse scalene triangle.

Sides: a = 35   b = 9.69771987616   c = 42

Area: T = 128.1565543252
Perimeter: p = 86.69771987616
Semiperimeter: s = 43.34985993808

Angle ∠ A = α = 39° = 0.68106784083 rad
Angle ∠ B = β = 10.04114908974° = 10°2'29″ = 0.1755257078 rad
Angle ∠ C = γ = 130.9598509103° = 130°57'31″ = 2.28656571673 rad

Height: ha = 7.32331739001
Height: hb = 26.43114564241
Height: hc = 6.10326449167

Median: ma = 24.95553167063
Median: mb = 38.35435015878
Median: mc = 14.78223486602

Inradius: r = 2.95663940954
Circumradius: R = 27.80877752587

Vertex coordinates: A[42; 0] B[0; 0] C[34.4643861145; 6.10326449167]
Centroid: CG[25.4887953715; 2.03442149722]
Coordinates of the circumscribed circle: U[21; -18.22883396072]
Coordinates of the inscribed circle: I[33.65114006192; 2.95663940954]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141° = 0.68106784083 rad
∠ B' = β' = 169.9598509103° = 169°57'31″ = 0.1755257078 rad
∠ C' = γ' = 49.04114908974° = 49°2'29″ = 2.28656571673 rad




How did we calculate this triangle?

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

a = 35 ; ; c = 42 ; ; alpha = 39° ; ;

2. From angle α, c and side a we calculate b - by using the law of cosines and quadratic equation:

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 35**2 = 42**2 + b**2 - 2 * 42 * b * cos(39° ) ; ; ; ; ; ; b**2 -65.28b +539 =0 ; ; a=1; b=-65.28; c=539 ; ; D = b**2 - 4ac = 65.28**2 - 4 * 1 * 539 = 2105.51244521 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 65.28 ± sqrt{ 2105.51 } }{ 2 } ; ; b_{1,2} = 32.64013038 ± 22.9429316196 ; ; b_{1} = 55.5830619996 ; ; b_{2} = 9.69719876041 ; ; ; ; (b -55.5830619996) (b -9.69719876041) = 0 ; ;
 ; ; b > 0 ; ; ; ; b = 55.583 ; ;


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 = 35 ; ; b = 9.7 ; ; c = 42 ; ;

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

p = a+b+c = 35+9.7+42 = 86.7 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 86.7 }{ 2 } = 43.35 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43.35 * (43.35-35)(43.35-9.7)(43.35-42) } ; ; T = sqrt{ 16423.84 } = 128.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 128.16 }{ 35 } = 7.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 128.16 }{ 9.7 } = 26.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 128.16 }{ 42 } = 6.1 ; ;

7. 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{ 35**2-9.7**2-42**2 }{ 2 * 9.7 * 42 } ) = 39° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.7**2-35**2-42**2 }{ 2 * 35 * 42 } ) = 10° 2'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42**2-35**2-9.7**2 }{ 2 * 9.7 * 35 } ) = 130° 57'31" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 128.16 }{ 43.35 } = 2.96 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 35 }{ 2 * sin 39° } = 27.81 ; ;





#2 Obtuse scalene triangle.

Sides: a = 35   b = 55.58330620008   c = 42

Area: T = 734.5710640596
Perimeter: p = 132.5833062001
Semiperimeter: s = 66.29215310004

Angle ∠ A = α = 39° = 0.68106784083 rad
Angle ∠ B = β = 91.95985091026° = 91°57'31″ = 1.60549787591 rad
Angle ∠ C = γ = 49.04114908974° = 49°2'29″ = 0.85659354862 rad

Height: ha = 41.97554651769
Height: hb = 26.43114564241
Height: hc = 34.98795543141

Median: ma = 46.04987610115
Median: mb = 26.87224915974
Median: mc = 41.42875076573

Inradius: r = 11.08109122902
Circumradius: R = 27.80877752587

Vertex coordinates: A[42; 0] B[0; 0] C[-1.19661521593; 34.98795543141]
Centroid: CG[13.60112826136; 11.6659851438]
Coordinates of the circumscribed circle: U[21; 18.22883396072]
Coordinates of the inscribed circle: I[10.70884689996; 11.08109122902]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141° = 0.68106784083 rad
∠ B' = β' = 88.04114908974° = 88°2'29″ = 1.60549787591 rad
∠ C' = γ' = 130.9598509103° = 130°57'31″ = 0.85659354862 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 35 ; ; c = 42 ; ; alpha = 39° ; ; : Nr. 1

2. From angle α, c and side a we calculate b - by using the law of cosines and quadratic equation:

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 35**2 = 42**2 + b**2 - 2 * 42 * b * cos(39° ) ; ; ; ; ; ; b**2 -65.28b +539 =0 ; ; a=1; b=-65.28; c=539 ; ; D = b**2 - 4ac = 65.28**2 - 4 * 1 * 539 = 2105.51244521 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 65.28 ± sqrt{ 2105.51 } }{ 2 } ; ; b_{1,2} = 32.64013038 ± 22.9429316196 ; ; b_{1} = 55.5830619996 ; ; b_{2} = 9.69719876041 ; ; ; ; (b -55.5830619996) (b -9.69719876041) = 0 ; ; : Nr. 1
 ; ; b > 0 ; ; ; ; b = 55.583 ; ; : Nr. 1


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 = 35 ; ; b = 55.58 ; ; c = 42 ; ;

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

p = a+b+c = 35+55.58+42 = 132.58 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 132.58 }{ 2 } = 66.29 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 66.29 * (66.29-35)(66.29-55.58)(66.29-42) } ; ; T = sqrt{ 539594.03 } = 734.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 734.57 }{ 35 } = 41.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 734.57 }{ 55.58 } = 26.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 734.57 }{ 42 } = 34.98 ; ;

7. 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{ 35**2-55.58**2-42**2 }{ 2 * 55.58 * 42 } ) = 39° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 55.58**2-35**2-42**2 }{ 2 * 35 * 42 } ) = 91° 57'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42**2-35**2-55.58**2 }{ 2 * 55.58 * 35 } ) = 49° 2'29" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 734.57 }{ 66.29 } = 11.08 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 35 }{ 2 * sin 39° } = 27.81 ; ; : Nr. 1




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