Triangle calculator

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

You have entered side a, b and area S.

Triangle has two solutions: a=8.94; b=7.5; c=14.99217714647 and a=8.94; b=7.5; c=6.8998839638.

#1 Obtuse scalene triangle.

Sides: a = 8.94   b = 7.5   c = 14.99217714647

Area: T = 25.17
Perimeter: p = 31.43217714647
Semiperimeter: s = 15.71658857323

Angle ∠ A = α = 26.59769998929° = 26°35'49″ = 0.46442052193 rad
Angle ∠ B = β = 22.06112492706° = 22°3'40″ = 0.38550414369 rad
Angle ∠ C = γ = 131.3421750837° = 131°20'30″ = 2.29223459974 rad

Height: ha = 5.63108724832
Height: hb = 6.712
Height: hc = 3.35878420081

Median: ma = 10.9788192284
Median: mb = 11.75990775924
Median: mc = 3.4499419819

Inradius: r = 1.60215642025
Circumradius: R = 9.98440909486

Vertex coordinates: A[14.99217714647; 0] B[0; 0] C[8.28554388567; 3.35878420081]
Centroid: CG[7.75990701071; 1.11992806694]
Coordinates of the circumscribed circle: U[7.49658857323; -6.59549806033]
Coordinates of the inscribed circle: I[8.21658857323; 1.60215642025]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4033000107° = 153°24'11″ = 0.46442052193 rad
∠ B' = β' = 157.9398750729° = 157°56'20″ = 0.38550414369 rad
∠ C' = γ' = 48.65882491635° = 48°39'30″ = 2.29223459974 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 = 8.94 ; ; b = 7.5 ; ; c = 14.99 ; ;

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

p = a+b+c = 8.94+7.5+14.99 = 31.43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.43 }{ 2 } = 15.72 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.72 * (15.72-8.94)(15.72-7.5)(15.72-14.99) } ; ; T = sqrt{ 633.53 } = 25.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.17 }{ 8.94 } = 5.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.17 }{ 7.5 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.17 }{ 14.99 } = 3.36 ; ;

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{ 8.94**2-7.5**2-14.99**2 }{ 2 * 7.5 * 14.99 } ) = 26° 35'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.5**2-8.94**2-14.99**2 }{ 2 * 8.94 * 14.99 } ) = 22° 3'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.99**2-8.94**2-7.5**2 }{ 2 * 7.5 * 8.94 } ) = 131° 20'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.17 }{ 15.72 } = 1.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.94 }{ 2 * sin 26° 35'49" } = 9.98 ; ;





#2 Acute scalene triangle.

Sides: a = 8.94   b = 7.5   c = 6.8998839638

Area: T = 25.17
Perimeter: p = 23.3398839638
Semiperimeter: s = 11.6699419819

Angle ∠ A = α = 76.63549934795° = 76°38'6″ = 1.33875329585 rad
Angle ∠ B = β = 54.7076757357° = 54°42'24″ = 0.9554813039 rad
Angle ∠ C = γ = 48.65882491635° = 48°39'30″ = 0.84992466562 rad

Height: ha = 5.63108724832
Height: hb = 6.712
Height: hc = 7.29768792785

Median: ma = 5.65216452627
Median: mb = 7.05495598568
Median: mc = 7.49658857323

Inradius: r = 2.15769195719
Circumradius: R = 4.59444298542

Vertex coordinates: A[6.8998839638; 0] B[0; 0] C[5.16551866176; 7.29768792785]
Centroid: CG[4.02113420852; 2.43222930928]
Coordinates of the circumscribed circle: U[3.4499419819; 3.03548457289]
Coordinates of the inscribed circle: I[4.1699419819; 2.15769195719]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.365500652° = 103°21'54″ = 1.33875329585 rad
∠ B' = β' = 125.2933242643° = 125°17'36″ = 0.9554813039 rad
∠ C' = γ' = 131.3421750837° = 131°20'30″ = 0.84992466562 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 = 8.94 ; ; b = 7.5 ; ; c = 6.9 ; ;

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

p = a+b+c = 8.94+7.5+6.9 = 23.34 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.34 }{ 2 } = 11.67 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.67 * (11.67-8.94)(11.67-7.5)(11.67-6.9) } ; ; T = sqrt{ 633.53 } = 25.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.17 }{ 8.94 } = 5.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.17 }{ 7.5 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.17 }{ 6.9 } = 7.3 ; ;

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{ 8.94**2-7.5**2-6.9**2 }{ 2 * 7.5 * 6.9 } ) = 76° 38'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.5**2-8.94**2-6.9**2 }{ 2 * 8.94 * 6.9 } ) = 54° 42'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.9**2-8.94**2-7.5**2 }{ 2 * 7.5 * 8.94 } ) = 48° 39'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.17 }{ 11.67 } = 2.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.94 }{ 2 * sin 76° 38'6" } = 4.59 ; ;




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