Triangle calculator SSA

Please enter two sides and a non-included angle
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Triangle has two solutions with side c=128.1954660419 and with side c=59.16878309783

#1 Acute scalene triangle.

Sides: a = 113   b = 72   c = 128.1954660419

Area: T = 4050.233325686
Perimeter: p = 313.1954660419
Semiperimeter: s = 156.597733021

Angle ∠ A = α = 61.35768472112° = 61°21'25″ = 1.07108790025 rad
Angle ∠ B = β = 34° = 0.59334119457 rad
Angle ∠ C = γ = 84.64331527888° = 84°38'35″ = 1.47773017054 rad

Height: ha = 71.68655443692
Height: hb = 112.5066479357
Height: hc = 63.18987980922

Median: ma = 87.27436241942
Median: mb = 115.3499189334
Median: mc = 69.77112853544

Inradius: r = 25.86439994146
Circumradius: R = 64.3788499399

Vertex coordinates: A[128.1954660419; 0] B[0; 0] C[93.68112456987; 63.18987980922]
Centroid: CG[73.95986353726; 21.06329326974]
Coordinates of the circumscribed circle: U[64.09773302096; 6.01102782688]
Coordinates of the inscribed circle: I[84.59773302096; 25.86439994146]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 118.6433152789° = 118°38'35″ = 1.07108790025 rad
∠ B' = β' = 146° = 0.59334119457 rad
∠ C' = γ' = 95.35768472112° = 95°21'25″ = 1.47773017054 rad


How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is such that we try to calculate all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase is the calculation of other characteristics of the triangle, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Use the Law of Cosines

a=113 b=72 β=34  b2=a2+c22accosβ 722=1132+c22 113 c cos(34)  c2187.362c+7585=0  p=1;q=187.362;r=7585 D=q24pr=187.3622417585=4764.70318266 D>0  c1,2=q±D2p=187.36±4764.72 c1,2=93.6812457±34.5134147204 c1=128.194660419 c2=59.1678309783   Factored form of the equation:  (c128.194660419)(c59.1678309783)=0   c>0a = 113 \ \\ b = 72 \ \\ β = 34^\circ \ \\ \ \\ b^2 = a^2 + c^2 - 2ac \cos β \ \\ 72^2 = 113^2 + c^2 -2 \cdot \ 113 \cdot \ c \cdot \ \cos (34^\circ ) \ \\ \ \\ c^2 -187.362c +7585 =0 \ \\ \ \\ p=1; q=-187.362; r=7585 \ \\ D = q^2 - 4pr = 187.362^2 - 4\cdot 1 \cdot 7585 = 4764.70318266 \ \\ D>0 \ \\ \ \\ c_{1,2} = \dfrac{ -q \pm \sqrt{ D } }{ 2p } = \dfrac{ 187.36 \pm \sqrt{ 4764.7 } }{ 2 } \ \\ c_{1,2} = 93.6812457 \pm 34.5134147204 \ \\ c_{1} = 128.194660419 \ \\ c_{2} = 59.1678309783 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (c -128.194660419) (c -59.1678309783) = 0 \ \\ \ \\ \ \\ c>0

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=113 b=72 c=128.19a = 113 \ \\ b = 72 \ \\ c = 128.19

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

p=a+b+c=113+72+128.19=313.19p = a+b+c = 113+72+128.19 = 313.19

3. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles that it is given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s=p2=313.192=156.6s = \dfrac{ p }{ 2 } = \dfrac{ 313.19 }{ 2 } = 156.6

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides are known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

T=s(sa)(sb)(sc) T=156.6(156.6113)(156.672)(156.6128.19) T=16404389.43=4050.23T = \sqrt{ s(s-a)(s-b)(s-c) } \ \\ T = \sqrt{ 156.6(156.6-113)(156.6-72)(156.6-128.19) } \ \\ T = \sqrt{ 16404389.43 } = 4050.23

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The area of a triangle is half of the product of the length of the base and the height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

T=aha2  ha=2 Ta=2 4050.23113=71.69 hb=2 Tb=2 4050.2372=112.51 hc=2 Tc=2 4050.23128.19=63.19T = \dfrac{ a h _a }{ 2 } \ \\ \ \\ h _a = \dfrac{ 2 \ T }{ a } = \dfrac{ 2 \cdot \ 4050.23 }{ 113 } = 71.69 \ \\ h _b = \dfrac{ 2 \ T }{ b } = \dfrac{ 2 \cdot \ 4050.23 }{ 72 } = 112.51 \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 4050.23 }{ 128.19 } = 63.19

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding the angles of a triangle when we know all three sides. The cosine rule, also known as the law of cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines is the extrapolation of the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines, because cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from cosine value.

a2=b2+c22bccosα  α=arccos(b2+c2a22bc)=arccos(722+128.19211322 72 128.19)=612125"  b2=a2+c22accosβ β=arccos(a2+c2b22ac)=arccos(1132+128.1927222 113 128.19)=34 γ=180αβ=180612125"34=843835"a^2 = b^2+c^2 - 2bc \cos α \ \\ \ \\ α = \arccos(\dfrac{ b^2+c^2-a^2 }{ 2bc } ) = \arccos(\dfrac{ 72^2+128.19^2-113^2 }{ 2 \cdot \ 72 \cdot \ 128.19 } ) = 61^\circ 21'25" \ \\ \ \\ b^2 = a^2+c^2 - 2ac \cos β \ \\ β = \arccos(\dfrac{ a^2+c^2-b^2 }{ 2ac } ) = \arccos(\dfrac{ 113^2+128.19^2-72^2 }{ 2 \cdot \ 113 \cdot \ 128.19 } ) = 34^\circ \ \\ γ = 180^\circ - α - β = 180^\circ - 61^\circ 21'25" - 34^\circ = 84^\circ 38'35"

7. Inradius

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

T=rs r=Ts=4050.23156.6=25.86T = rs \ \\ r = \dfrac{ T }{ s } = \dfrac{ 4050.23 }{ 156.6 } = 25.86

8. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R=abc4 rs=113 72 128.194 25.864 156.597=64.38R = \dfrac{ a b c }{ 4 \ r s } = \dfrac{ 113 \cdot \ 72 \cdot \ 128.19 }{ 4 \cdot \ 25.864 \cdot \ 156.597 } = 64.38

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.

ma=2b2+2c2a22=2 722+2 128.19211322=87.274 mb=2c2+2a2b22=2 128.192+2 11327222=115.349 mc=2a2+2b2c22=2 1132+2 722128.1922=69.771m_a = \dfrac{ \sqrt{ 2b^2+2c^2 - a^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 72^2+2 \cdot \ 128.19^2 - 113^2 } }{ 2 } = 87.274 \ \\ m_b = \dfrac{ \sqrt{ 2c^2+2a^2 - b^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 128.19^2+2 \cdot \ 113^2 - 72^2 } }{ 2 } = 115.349 \ \\ m_c = \dfrac{ \sqrt{ 2a^2+2b^2 - c^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 113^2+2 \cdot \ 72^2 - 128.19^2 } }{ 2 } = 69.771



#2 Obtuse scalene triangle.

Sides: a = 113   b = 72   c = 59.16878309783

Area: T = 1869.372206262
Perimeter: p = 244.1687830978
Semiperimeter: s = 122.0843915489

Angle ∠ A = α = 118.6433152789° = 118°38'35″ = 2.07107136511 rad
Angle ∠ B = β = 34° = 0.59334119457 rad
Angle ∠ C = γ = 27.35768472112° = 27°21'25″ = 0.47774670568 rad

Height: ha = 33.08662311968
Height: hb = 51.92770017395
Height: hc = 63.18987980922

Median: ma = 33.91440990053
Median: mb = 82.69877394572
Median: mc = 90.00771771823

Inradius: r = 15.31221896126
Circumradius: R = 64.3788499399

Vertex coordinates: A[59.16878309783; 0] B[0; 0] C[93.68112456987; 63.18987980922]
Centroid: CG[50.95496922257; 21.06329326974]
Coordinates of the circumscribed circle: U[29.58439154891; 57.17985198234]
Coordinates of the inscribed circle: I[50.08439154891; 15.31221896126]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 61.35768472112° = 61°21'25″ = 2.07107136511 rad
∠ B' = β' = 146° = 0.59334119457 rad
∠ C' = γ' = 152.6433152789° = 152°38'35″ = 0.47774670568 rad

Calculate another triangle

How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is such that we try to calculate all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase is the calculation of other characteristics of the triangle, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Use the Law of Cosines

a=113 b=72 β=34  b2=a2+c22accosβ 722=1132+c22 113 c cos(34)  c2187.362c+7585=0  p=1;q=187.362;r=7585 D=q24pr=187.3622417585=4764.70318266 D>0  c1,2=q±D2p=187.36±4764.72 c1,2=93.6812457±34.5134147204 c1=128.194660419 c2=59.1678309783   Factored form of the equation:  (c128.194660419)(c59.1678309783)=0   c>0a = 113 \ \\ b = 72 \ \\ β = 34^\circ \ \\ \ \\ b^2 = a^2 + c^2 - 2ac \cos β \ \\ 72^2 = 113^2 + c^2 -2 \cdot \ 113 \cdot \ c \cdot \ \cos (34^\circ ) \ \\ \ \\ c^2 -187.362c +7585 =0 \ \\ \ \\ p=1; q=-187.362; r=7585 \ \\ D = q^2 - 4pr = 187.362^2 - 4\cdot 1 \cdot 7585 = 4764.70318266 \ \\ D>0 \ \\ \ \\ c_{1,2} = \dfrac{ -q \pm \sqrt{ D } }{ 2p } = \dfrac{ 187.36 \pm \sqrt{ 4764.7 } }{ 2 } \ \\ c_{1,2} = 93.6812457 \pm 34.5134147204 \ \\ c_{1} = 128.194660419 \ \\ c_{2} = 59.1678309783 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (c -128.194660419) (c -59.1678309783) = 0 \ \\ \ \\ \ \\ c>0

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=113 b=72 c=59.17a = 113 \ \\ b = 72 \ \\ c = 59.17

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

p=a+b+c=113+72+59.17=244.17p = a+b+c = 113+72+59.17 = 244.17

3. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles that it is given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s=p2=244.172=122.08s = \dfrac{ p }{ 2 } = \dfrac{ 244.17 }{ 2 } = 122.08

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides are known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

T=s(sa)(sb)(sc) T=122.08(122.08113)(122.0872)(122.0859.17) T=3494551.91=1869.37T = \sqrt{ s(s-a)(s-b)(s-c) } \ \\ T = \sqrt{ 122.08(122.08-113)(122.08-72)(122.08-59.17) } \ \\ T = \sqrt{ 3494551.91 } = 1869.37

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The area of a triangle is half of the product of the length of the base and the height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

T=aha2  ha=2 Ta=2 1869.37113=33.09 hb=2 Tb=2 1869.3772=51.93 hc=2 Tc=2 1869.3759.17=63.19T = \dfrac{ a h _a }{ 2 } \ \\ \ \\ h _a = \dfrac{ 2 \ T }{ a } = \dfrac{ 2 \cdot \ 1869.37 }{ 113 } = 33.09 \ \\ h _b = \dfrac{ 2 \ T }{ b } = \dfrac{ 2 \cdot \ 1869.37 }{ 72 } = 51.93 \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 1869.37 }{ 59.17 } = 63.19

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding the angles of a triangle when we know all three sides. The cosine rule, also known as the law of cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines is the extrapolation of the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines, because cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from cosine value.

a2=b2+c22bccosα  α=arccos(b2+c2a22bc)=arccos(722+59.17211322 72 59.17)=1183835"  b2=a2+c22accosβ β=arccos(a2+c2b22ac)=arccos(1132+59.1727222 113 59.17)=34 γ=180αβ=1801183835"34=272125"a^2 = b^2+c^2 - 2bc \cos α \ \\ \ \\ α = \arccos(\dfrac{ b^2+c^2-a^2 }{ 2bc } ) = \arccos(\dfrac{ 72^2+59.17^2-113^2 }{ 2 \cdot \ 72 \cdot \ 59.17 } ) = 118^\circ 38'35" \ \\ \ \\ b^2 = a^2+c^2 - 2ac \cos β \ \\ β = \arccos(\dfrac{ a^2+c^2-b^2 }{ 2ac } ) = \arccos(\dfrac{ 113^2+59.17^2-72^2 }{ 2 \cdot \ 113 \cdot \ 59.17 } ) = 34^\circ \ \\ γ = 180^\circ - α - β = 180^\circ - 118^\circ 38'35" - 34^\circ = 27^\circ 21'25"

7. Inradius

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

T=rs r=Ts=1869.37122.08=15.31T = rs \ \\ r = \dfrac{ T }{ s } = \dfrac{ 1869.37 }{ 122.08 } = 15.31

8. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R=abc4 rs=113 72 59.174 15.312 122.084=64.38R = \dfrac{ a b c }{ 4 \ r s } = \dfrac{ 113 \cdot \ 72 \cdot \ 59.17 }{ 4 \cdot \ 15.312 \cdot \ 122.084 } = 64.38

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.

ma=2b2+2c2a22=2 722+2 59.17211322=33.914 mb=2c2+2a2b22=2 59.172+2 11327222=82.698 mc=2a2+2b2c22=2 1132+2 72259.1722=90.007m_a = \dfrac{ \sqrt{ 2b^2+2c^2 - a^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 72^2+2 \cdot \ 59.17^2 - 113^2 } }{ 2 } = 33.914 \ \\ m_b = \dfrac{ \sqrt{ 2c^2+2a^2 - b^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 59.17^2+2 \cdot \ 113^2 - 72^2 } }{ 2 } = 82.698 \ \\ m_c = \dfrac{ \sqrt{ 2a^2+2b^2 - c^2 } }{ 2 } = \dfrac{ \sqrt{ 2 \cdot \ 113^2+2 \cdot \ 72^2 - 59.17^2 } }{ 2 } = 90.007

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