SSA Triangle Calculator (Ambiguous Case)

Q: Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Q: Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Q: Does it run in my browser?

Yes. Your values are processed by code running in your browser tab, on your own device.

Q: Is my data uploaded?

No. Your values are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

Q: Why does SSA sometimes give two triangles?

Because the sine of an angle and its supplement are equal, so angle B can be acute or obtuse. If the obtuse one fits with A, there are two triangles.

Q: When does SSA give no triangle?

When the side opposite the angle is shorter than b sin A: it is too short to reach the base line.

Q: Is SSA a congruence rule?

No. Unlike SSS, SAS, ASA and AAS, SSA does not always fix one triangle, which is exactly the ambiguous case.

Solving the triangle: solution 1

1Law of sines for B
$\sin B=\dfrac{b\sin A}{a}$
$\sin B=\dfrac{8\sin 30^\circ}{6}=0.666667$
2Take the acute angle
$B_1=\arcsin(\sin B)$
$B=41.8103^\circ$
3Angles add to a straight angle
$C=180^\circ-A-B$
$C=180^\circ-30^\circ-41.8103^\circ=108.1897^\circ$
4Law of sines for c
$\dfrac{c}{\sin C}=\dfrac{a}{\sin A}$
$c=\dfrac{6\sin 108.1897^\circ}{\sin 30^\circ}=11.4003$

Area, heights and circles: solution 1

1Semi-perimeter
$s=\dfrac{a+b+c}{2}$
$s=\dfrac{6+8+11.4003}{2}=12.7002$
2Heron's formula
$K=\sqrt{s(s-a)(s-b)(s-c)}$
$K=\sqrt{12.7002(6.7002)(4.7002)(1.2998)}=22.8007$
3Area from two sides and the angle between them
$K=\tfrac12 ab\sin C$
$K=\tfrac12\cdot 6\cdot 8\cdot \sin(108.1897^\circ)=22.8007$
Both methods give the same area.
4Heights
$h_a=\dfrac{2K}{a},\; h_b=\dfrac{2K}{b},\; h_c=\dfrac{2K}{c}$
$h_a=7.6002,\; h_b=5.7002,\; h_c=4$
5Medians
$m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}$
$m_a=9.38,\; m_b=8.1844,\; m_c=4.1843$
6Inscribed circle
$r=\dfrac{K}{s}$
$r=\dfrac{22.8007}{12.7002}=1.7953$
7Circumscribed circle
$R=\dfrac{abc}{4K}$
$R=\dfrac{6\cdot 8\cdot 11.4003}{4\cdot 22.8007}=6$

Solving the triangle: solution 2

1Law of sines for B
$\sin B=\dfrac{b\sin A}{a}$
$\sin B=\dfrac{8\sin 30^\circ}{6}=0.666667$
2Take the obtuse angle
$B_2=180^\circ-B_1$
$B=180^\circ-41.8103^\circ=138.1897^\circ$
3Angles add to a straight angle
$C=180^\circ-A-B$
$C=180^\circ-30^\circ-138.1897^\circ=11.8103^\circ$
4Law of sines for c
$\dfrac{c}{\sin C}=\dfrac{a}{\sin A}$
$c=\dfrac{6\sin 11.8103^\circ}{\sin 30^\circ}=2.4561$

Area, heights and circles: solution 2

1Semi-perimeter
$s=\dfrac{a+b+c}{2}$
$s=\dfrac{6+8+2.4561}{2}=8.228$
2Heron's formula
$K=\sqrt{s(s-a)(s-b)(s-c)}$
$K=\sqrt{8.228(2.228)(0.228)(5.772)}=4.9121$
3Area from two sides and the angle between them
$K=\tfrac12 ab\sin C$
$K=\tfrac12\cdot 6\cdot 8\cdot \sin(11.8103^\circ)=4.9121$
Both methods give the same area.
4Heights
$h_a=\dfrac{2K}{a},\; h_b=\dfrac{2K}{b},\; h_c=\dfrac{2K}{c}$
$h_a=1.6374,\; h_b=1.228,\; h_c=4$
5Medians
$m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}$
$m_a=5.1006,\; m_b=2.2397,\; m_c=6.9636$
6Inscribed circle
$r=\dfrac{K}{s}$
$r=\dfrac{4.9121}{8.228}=0.597$
7Circumscribed circle
$R=\dfrac{abc}{4K}$
$R=\dfrac{6\cdot 8\cdot 2.4561}{4\cdot 4.9121}=6$

About SSA Triangle Calculator

This SSA triangle calculator handles the ambiguous case of the law of sines. Enter side a, side b and angle A, and it reports whether there are two triangles, one triangle or none, with a diagram and working for each. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The deciding quantity is the height h = b sin A. When a is shorter than b, three things can happen: a is less than h and no triangle exists, a equals h and a right triangle results, or a is between h and b and two triangles fit.

In the two-triangle case the angle B is either acute or obtuse, because sin B = sin(180° − B). Both are valid when the obtuse angle still leaves room for angle A.

How to use SSA Triangle Calculator

Pick the mode that matches what you know: SSS, SAS, ASA, AAS, SSA or Right triangle.
Type the three values in the boxes. Sides a, b, c sit opposite angles A, B, C. In right mode fill exactly two boxes.
Choose a length unit and whether angles are in degrees or radians. Changing the angle unit converts the numbers you already typed.
Read the sides, angles, area and other measures, and compare them with the scaled diagram.
Open the step-by-step working to see each law of cosines or law of sines line, then Heron’s formula for the area.
In SSA mode, check whether one solution, two solutions or none appear, and read the note that explains why.

Features

Six modes: SSS, SAS, ASA, AAS, SSA and right triangle.
Both solutions shown in the ambiguous SSA case, each with a diagram and its own working.
Area by Heron’s formula and by one half a b sin C, which agree.
Perimeter, three heights, three medians, inradius and circumradius.
Triangle type by angles (acute, right, obtuse) and by sides (scalene, isosceles, equilateral).
A scaled, labelled SVG diagram that follows light and dark mode.
Degrees or radians, and eight length units from millimetres to miles.
Clear messages when the values cannot form a triangle, instead of a wrong answer.
Worked examples and KaTeX formulas for every step.

The SSA test

Find sin B first, then check the cases.

sin B = b sin A / a
h = b sin A
B₁ = arcsin(sin B),  B₂ = 180° − B₁

Worked examples

a = 6, b = 8, A = 30° gives Two triangles: c = 11.4003 or 2.4561
Height h = 8 sin 30° = 4, and 4 < 6 < 8.
a = 9, b = 8, A = 30° gives One triangle: c = 14.9905
Side a is longer than b, so only the acute angle B works.

Privacy: how it works

Everything is calculated by code running in your browser tab. The values you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

Check the labelling: side a is opposite angle A. Swapping a side with its neighbour gives a different triangle.
In SAS the angle must be the one between the two sides. If it is not, use the SSA mode.
If two angles already add to 180° or more, no triangle exists. The calculator says so.
For a right triangle you can also use SSS with the sides 3, 4 and 5 and read C = 90°.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your values are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your values are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

Why does SSA sometimes give two triangles?

Because the sine of an angle and its supplement are equal, so angle B can be acute or obtuse. If the obtuse one fits with A, there are two triangles.

When does SSA give no triangle?

When the side opposite the angle is shorter than b sin A: it is too short to reach the base line.

Is SSA a congruence rule?

No. Unlike SSS, SAS, ASA and AAS, SSA does not always fix one triangle, which is exactly the ambiguous case.

Solving the triangle: solution 1

1Law of sines for B
$\sin B=\dfrac{b\sin A}{a}$
$\sin B=\dfrac{8\sin 30^\circ}{6}=0.666667$
2Take the acute angle
$B_1=\arcsin(\sin B)$
$B=41.8103^\circ$
3Angles add to a straight angle
$C=180^\circ-A-B$
$C=180^\circ-30^\circ-41.8103^\circ=108.1897^\circ$
4Law of sines for c
$\dfrac{c}{\sin C}=\dfrac{a}{\sin A}$
$c=\dfrac{6\sin 108.1897^\circ}{\sin 30^\circ}=11.4003$

Area, heights and circles: solution 1

1Semi-perimeter
$s=\dfrac{a+b+c}{2}$
$s=\dfrac{6+8+11.4003}{2}=12.7002$
2Heron's formula
$K=\sqrt{s(s-a)(s-b)(s-c)}$
$K=\sqrt{12.7002(6.7002)(4.7002)(1.2998)}=22.8007$
3Area from two sides and the angle between them
$K=\tfrac12 ab\sin C$
$K=\tfrac12\cdot 6\cdot 8\cdot \sin(108.1897^\circ)=22.8007$
Both methods give the same area.
4Heights
$h_a=\dfrac{2K}{a},\; h_b=\dfrac{2K}{b},\; h_c=\dfrac{2K}{c}$
$h_a=7.6002,\; h_b=5.7002,\; h_c=4$
5Medians
$m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}$
$m_a=9.38,\; m_b=8.1844,\; m_c=4.1843$
6Inscribed circle
$r=\dfrac{K}{s}$
$r=\dfrac{22.8007}{12.7002}=1.7953$
7Circumscribed circle
$R=\dfrac{abc}{4K}$
$R=\dfrac{6\cdot 8\cdot 11.4003}{4\cdot 22.8007}=6$

Solving the triangle: solution 2

1Law of sines for B
$\sin B=\dfrac{b\sin A}{a}$
$\sin B=\dfrac{8\sin 30^\circ}{6}=0.666667$
2Take the obtuse angle
$B_2=180^\circ-B_1$
$B=180^\circ-41.8103^\circ=138.1897^\circ$
3Angles add to a straight angle
$C=180^\circ-A-B$
$C=180^\circ-30^\circ-138.1897^\circ=11.8103^\circ$
4Law of sines for c
$\dfrac{c}{\sin C}=\dfrac{a}{\sin A}$
$c=\dfrac{6\sin 11.8103^\circ}{\sin 30^\circ}=2.4561$

Area, heights and circles: solution 2

1Semi-perimeter
$s=\dfrac{a+b+c}{2}$
$s=\dfrac{6+8+2.4561}{2}=8.228$
2Heron's formula
$K=\sqrt{s(s-a)(s-b)(s-c)}$
$K=\sqrt{8.228(2.228)(0.228)(5.772)}=4.9121$
3Area from two sides and the angle between them
$K=\tfrac12 ab\sin C$
$K=\tfrac12\cdot 6\cdot 8\cdot \sin(11.8103^\circ)=4.9121$
Both methods give the same area.
4Heights
$h_a=\dfrac{2K}{a},\; h_b=\dfrac{2K}{b},\; h_c=\dfrac{2K}{c}$
$h_a=1.6374,\; h_b=1.228,\; h_c=4$
5Medians
$m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}$
$m_a=5.1006,\; m_b=2.2397,\; m_c=6.9636$
6Inscribed circle
$r=\dfrac{K}{s}$
$r=\dfrac{4.9121}{8.228}=0.597$
7Circumscribed circle
$R=\dfrac{abc}{4K}$
$R=\dfrac{6\cdot 8\cdot 2.4561}{4\cdot 4.9121}=6$

About SSA Triangle Calculator

In the two-triangle case the angle B is either acute or obtuse, because sin B = sin(180° − B). Both are valid when the obtuse angle still leaves room for angle A.

SSA Triangle Calculator

Area of solution 1

Solution 1

Sides and angles

Measures

Solution 2

Sides and angles

Measures

Solving the triangle: solution 1

Area, heights and circles: solution 1

Solving the triangle: solution 2

Area, heights and circles: solution 2

About SSA Triangle Calculator

How to use SSA Triangle Calculator

Features

The SSA test

Worked examples

Privacy: how it works

Tips

Frequently asked questions

Is it really free?

Do I need an account?

Does it run in my browser?

Is my data uploaded?

Why does SSA sometimes give two triangles?

When does SSA give no triangle?

Is SSA a congruence rule?

More Triangle Calculator pages

SSA Triangle Calculator

Area of solution 1

Solution 1

Sides and angles

Measures

Solution 2

Sides and angles

Measures

Solving the triangle: solution 1

Area, heights and circles: solution 1

Solving the triangle: solution 2

Area, heights and circles: solution 2

About SSA Triangle Calculator

How to use SSA Triangle Calculator

Features

The SSA test

Worked examples

Privacy: how it works

Tips

Frequently asked questions

Is it really free?

Do I need an account?

Does it run in my browser?

Is my data uploaded?

Why does SSA sometimes give two triangles?

When does SSA give no triangle?

Is SSA a congruence rule?

More Triangle Calculator pages