Triangle calculator

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

You have entered side b, c and area T.

Triangle has two solutions: a=31.11774184616; b=21.9; c=10.4 and a=14.39660504404; b=21.9; c=10.4.

#1 Obtuse scalene triangle.

Sides: a = 31.11774184616   b = 21.9   c = 10.4

Area: T = 62.6
Perimeter: p = 63.41774184616
Semiperimeter: s = 31.70987092308

Angle ∠ A = α = 146.6533467075° = 146°39'12″ = 2.56595858599 rad
Angle ∠ B = β = 22.76600128159° = 22°45'36″ = 0.39772371614 rad
Angle ∠ C = γ = 10.58765201095° = 10°35'11″ = 0.18547696322 rad

Height: ha = 4.0233470011
Height: hb = 5.71768949772
Height: hc = 12.03884615385

Median: ma = 7.19880252202
Median: mb = 20.45329793883
Median: mc = 26.39990883528

Inradius: r = 1.97442210112
Circumradius: R = 28.30439266327

Vertex coordinates: A[10.4; 0] B[0; 0] C[28.69444101787; 12.03884615385]
Centroid: CG[13.03114700596; 4.01328205128]
Coordinates of the circumscribed circle: U[5.2; 27.8222154173]
Coordinates of the inscribed circle: I[9.80987092308; 1.97442210112]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 33.34765329254° = 33°20'48″ = 2.56595858599 rad
∠ B' = β' = 157.2439987184° = 157°14'24″ = 0.39772371614 rad
∠ C' = γ' = 169.413347989° = 169°24'49″ = 0.18547696322 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 = 31.12 ; ; b = 21.9 ; ; c = 10.4 ; ;

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

p = a+b+c = 31.12+21.9+10.4 = 63.42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63.42 }{ 2 } = 31.71 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.71 * (31.71-31.12)(31.71-21.9)(31.71-10.4) } ; ; T = sqrt{ 3918.76 } = 62.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.6 }{ 31.12 } = 4.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.6 }{ 21.9 } = 5.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.6 }{ 10.4 } = 12.04 ; ;

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{ 31.12**2-21.9**2-10.4**2 }{ 2 * 21.9 * 10.4 } ) = 146° 39'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.9**2-31.12**2-10.4**2 }{ 2 * 31.12 * 10.4 } ) = 22° 45'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.4**2-31.12**2-21.9**2 }{ 2 * 21.9 * 31.12 } ) = 10° 35'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.6 }{ 31.71 } = 1.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 31.12 }{ 2 * sin 146° 39'12" } = 28.3 ; ;





#2 Obtuse scalene triangle.

Sides: a = 14.39660504404   b = 21.9   c = 10.4

Area: T = 62.6
Perimeter: p = 46.69660504404
Semiperimeter: s = 23.34880252202

Angle ∠ A = α = 33.34765329254° = 33°20'48″ = 0.58220067937 rad
Angle ∠ B = β = 123.2555488315° = 123°15'20″ = 2.15112140922 rad
Angle ∠ C = γ = 23.39879787596° = 23°23'53″ = 0.40883717677 rad

Height: ha = 8.69768297672
Height: hb = 5.71768949772
Height: hc = 12.03884615385

Median: ma = 15.55987092308
Median: mb = 6.14882220309
Median: mc = 17.787730261

Inradius: r = 2.68111689387
Circumradius: R = 13.09444267105

Vertex coordinates: A[10.4; 0] B[0; 0] C[-7.89444101787; 12.03884615385]
Centroid: CG[0.83551966071; 4.01328205128]
Coordinates of the circumscribed circle: U[5.2; 12.01876541337]
Coordinates of the inscribed circle: I[1.44880252202; 2.68111689387]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.6533467075° = 146°39'12″ = 0.58220067937 rad
∠ B' = β' = 56.74545116851° = 56°44'40″ = 2.15112140922 rad
∠ C' = γ' = 156.602202124° = 156°36'7″ = 0.40883717677 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 = 14.4 ; ; b = 21.9 ; ; c = 10.4 ; ;

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

p = a+b+c = 14.4+21.9+10.4 = 46.7 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46.7 }{ 2 } = 23.35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.35 * (23.35-14.4)(23.35-21.9)(23.35-10.4) } ; ; T = sqrt{ 3918.76 } = 62.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.6 }{ 14.4 } = 8.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.6 }{ 21.9 } = 5.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.6 }{ 10.4 } = 12.04 ; ;

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{ 14.4**2-21.9**2-10.4**2 }{ 2 * 21.9 * 10.4 } ) = 33° 20'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.9**2-14.4**2-10.4**2 }{ 2 * 14.4 * 10.4 } ) = 123° 15'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.4**2-14.4**2-21.9**2 }{ 2 * 21.9 * 14.4 } ) = 23° 23'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.6 }{ 23.35 } = 2.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14.4 }{ 2 * sin 33° 20'48" } = 13.09 ; ;




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