Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=16.85; b=7.35; c=10.07993227815 and a=16.85; b=7.35; c=22.80990720959.

#1 Obtuse scalene triangle.

Sides: a = 16.85   b = 7.35   c = 10.07993227815

Area: T = 18.52443520017
Perimeter: p = 34.27993227815
Semiperimeter: s = 17.14396613907

Angle ∠ A = α = 149.9943576315° = 149°59'37″ = 2.61878817635 rad
Angle ∠ B = β = 12.6° = 12°36' = 0.22199114858 rad
Angle ∠ C = γ = 17.40664236849° = 17°24'23″ = 0.30437994043 rad

Height: ha = 2.19987361426
Height: hb = 5.04106400005
Height: hc = 3.67657136175

Median: ma = 2.61328526301
Median: mb = 13.38985024878
Median: mc = 11.98222499167

Inradius: r = 1.08107886795
Circumradius: R = 16.84767286746

Vertex coordinates: A[10.07993227815; 0] B[0; 0] C[16.44441974387; 3.67657136175]
Centroid: CG[8.84111734067; 1.22552378725]
Coordinates of the circumscribed circle: U[5.04396613907; 16.0755262987]
Coordinates of the inscribed circle: I[9.79896613907; 1.08107886795]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.00664236849° = 30°23″ = 2.61878817635 rad
∠ B' = β' = 167.4° = 167°24' = 0.22199114858 rad
∠ C' = γ' = 162.5943576315° = 162°35'37″ = 0.30437994043 rad




How did we calculate this triangle?

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 = 16.85 ; ; b = 7.35 ; ; c = 10.08 ; ;

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

p = a+b+c = 16.85+7.35+10.08 = 34.28 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34.28 }{ 2 } = 17.14 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.14 * (17.14-16.85)(17.14-7.35)(17.14-10.08) } ; ; T = sqrt{ 343.15 } = 18.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.52 }{ 16.85 } = 2.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.52 }{ 7.35 } = 5.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.52 }{ 10.08 } = 3.68 ; ;

5. 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{ 16.85**2-7.35**2-10.08**2 }{ 2 * 7.35 * 10.08 } ) = 149° 59'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.35**2-16.85**2-10.08**2 }{ 2 * 16.85 * 10.08 } ) = 12° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.08**2-16.85**2-7.35**2 }{ 2 * 7.35 * 16.85 } ) = 17° 24'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.52 }{ 17.14 } = 1.08 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.85 }{ 2 * sin 149° 59'37" } = 16.85 ; ;





#2 Obtuse scalene triangle.

Sides: a = 16.85   b = 7.35   c = 22.80990720959

Area: T = 41.9219808453
Perimeter: p = 47.00990720959
Semiperimeter: s = 23.50545360479

Angle ∠ A = α = 30.00664236849° = 30°23″ = 0.524371089 rad
Angle ∠ B = β = 12.6° = 12°36' = 0.22199114858 rad
Angle ∠ C = γ = 137.3943576315° = 137°23'37″ = 2.39879702778 rad

Height: ha = 4.97656449202
Height: hb = 11.40767505995
Height: hc = 3.67657136175

Median: ma = 14.70222960771
Median: mb = 19.71224962888
Median: mc = 6.23877125239

Inradius: r = 1.78334773836
Circumradius: R = 16.84767286746

Vertex coordinates: A[22.80990720959; 0] B[0; 0] C[16.44441974387; 3.67657136175]
Centroid: CG[13.08444231782; 1.22552378725]
Coordinates of the circumscribed circle: U[11.40545360479; -12.43995493695]
Coordinates of the inscribed circle: I[16.15545360479; 1.78334773836]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.9943576315° = 149°59'37″ = 0.524371089 rad
∠ B' = β' = 167.4° = 167°24' = 0.22199114858 rad
∠ C' = γ' = 42.60664236849° = 42°36'23″ = 2.39879702778 rad

Calculate another triangle

How did we calculate this triangle?

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 = 16.85 ; ; b = 7.35 ; ; c = 22.81 ; ;

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

p = a+b+c = 16.85+7.35+22.81 = 47.01 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47.01 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-16.85)(23.5-7.35)(23.5-22.81) } ; ; T = sqrt{ 1757.27 } = 41.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.92 }{ 16.85 } = 4.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.92 }{ 7.35 } = 11.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.92 }{ 22.81 } = 3.68 ; ;

5. 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{ 16.85**2-7.35**2-22.81**2 }{ 2 * 7.35 * 22.81 } ) = 30° 23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.35**2-16.85**2-22.81**2 }{ 2 * 16.85 * 22.81 } ) = 12° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22.81**2-16.85**2-7.35**2 }{ 2 * 7.35 * 16.85 } ) = 137° 23'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.92 }{ 23.5 } = 1.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.85 }{ 2 * sin 30° 23" } = 16.85 ; ;




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