Skip to content
ToolUlkaa Tool
All tools

Mathematics

Percentage CalculatorFraction CalculatorEquation SolverTensorMatrix CalculatorUnit ConverterNumber Base ConverterDerivative CalculatorIntegral CalculatorLimit CalculatorAlgebra CalculatorQuadratic Formula CalculatorFactoring CalculatorLogarithm CalculatorSquare Root CalculatorLong Division CalculatorSignificant Figures CalculatorLCM and GCF CalculatorPrime Factorization CalculatorTriangle CalculatorPythagorean Theorem CalculatorSlope CalculatorArea CalculatorVolume CalculatorScientific CalculatorGraphing Calculator

Categories

AcademicTextMathematicsStatisticsMachine LearningAIData ScienceFile
All tools
Mathematics

SAS Triangle Calculator

Enter two sides and the angle between them to find the third side and the other angles.

Known values

Two sides and the angle between them.

Try:

Sides a, b, c sit opposite angles A, B, C. In right mode the right angle is at C.

Area

24.1304 cm²
a = 7 cmb = 9 cmc = 7.001 cmA49.99°B80.01°C50°

Sides and angles

Side a
7 cm
Side b
9 cm
Side c
7.0006 cm
Angle A
49.9939°
Angle B
80.0061°
Angle C
50°

Measures

Area (Heron's formula)
24.1304 cm²
Area (½ ab sin C)
24.1304 cm²
Perimeter
23.0006 cm
Heights (on a, b, c)
6.8944, 5.3623, 6.8938 cm
Medians (to a, b, c)
7.2632, 5.3623, 7.2628 cm
Inradius r
2.0982 cm
Circumradius R
4.5693 cm
Type
acute, scalene

Solving the triangle: step by step

  1. 1Law of cosines for the third side
    c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos Cc2=a2+b2−2abcosC
    ↓
    c=72+92−2⋅7⋅9cos⁡50∘=7.0006c=\sqrt{7^2+9^2-2\cdot 7\cdot 9\cos 50^\circ}=7.0006c=72+92−2⋅7⋅9cos50∘​=7.0006
  2. 2Law of cosines for A
    cos⁡A=b2+c2−a22bc\cos A=\dfrac{b^2+c^2-a^2}{2bc}cosA=2bcb2+c2−a2​
    ↓
    A=arccos⁡ ⁣(92+7.00062−722⋅9⋅7.0006)=49.9939∘A=\arccos\!\left(\dfrac{9^2+7.0006^2-7^2}{2\cdot 9\cdot 7.0006}\right)=49.9939^\circA=arccos(2⋅9⋅7.000692+7.00062−72​)=49.9939∘
  3. 3Angles add to a straight angle
    B=180∘−A−CB=180^\circ-A-CB=180∘−A−C
    ↓
    B=180∘−49.9939∘−50∘=80.0061∘B=180^\circ-49.9939^\circ-50^\circ=80.0061^\circB=180∘−49.9939∘−50∘=80.0061∘

Area, heights and circles: step by step

  1. 1Semi-perimeter
    s=a+b+c2s=\dfrac{a+b+c}{2}s=2a+b+c​
    ↓
    s=7+9+7.00062=11.5003s=\dfrac{7+9+7.0006}{2}=11.5003s=27+9+7.0006​=11.5003
  2. 2Heron's formula
    K=s(s−a)(s−b)(s−c)K=\sqrt{s(s-a)(s-b)(s-c)}K=s(s−a)(s−b)(s−c)​
    ↓
    K=11.5003(4.5003)(2.5003)(4.4997)=24.1304K=\sqrt{11.5003(4.5003)(2.5003)(4.4997)}=24.1304K=11.5003(4.5003)(2.5003)(4.4997)​=24.1304
  3. 3Area from two sides and the angle between them
    K=12absin⁡CK=\tfrac12 ab\sin CK=21​absinC
    ↓
    K=12⋅7⋅9⋅sin⁡(50∘)=24.1304K=\tfrac12\cdot 7\cdot 9\cdot \sin(50^\circ)=24.1304K=21​⋅7⋅9⋅sin(50∘)=24.1304

    Both methods give the same area.

  4. 4Heights
    ha=2Ka,  hb=2Kb,  hc=2Kch_a=\dfrac{2K}{a},\; h_b=\dfrac{2K}{b},\; h_c=\dfrac{2K}{c}ha​=a2K​,hb​=b2K​,hc​=c2K​
    ↓
    ha=6.8944,  hb=5.3623,  hc=6.8938h_a=6.8944,\; h_b=5.3623,\; h_c=6.8938ha​=6.8944,hb​=5.3623,hc​=6.8938
  5. 5Medians
    ma=122b2+2c2−a2m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}ma​=21​2b2+2c2−a2​
    ↓
    ma=7.2632,  mb=5.3623,  mc=7.2628m_a=7.2632,\; m_b=5.3623,\; m_c=7.2628ma​=7.2632,mb​=5.3623,mc​=7.2628
  6. 6Inscribed circle
    r=Ksr=\dfrac{K}{s}r=sK​
    ↓
    r=24.130411.5003=2.0982r=\dfrac{24.1304}{11.5003}=2.0982r=11.500324.1304​=2.0982
  7. 7Circumscribed circle
    R=abc4KR=\dfrac{abc}{4K}R=4Kabc​
    ↓
    R=7⋅9⋅7.00064⋅24.1304=4.5693R=\dfrac{7\cdot 9\cdot 7.0006}{4\cdot 24.1304}=4.5693R=4⋅24.13047⋅9⋅7.0006​=4.5693
  1. Tools
  2. Mathematics
  3. Triangle Calculator
  4. SAS Triangle Calculator

About SAS Triangle Calculator

This SAS triangle calculator solves a triangle from two sides and the angle between them. It finds the third side with the law of cosines, then the remaining angles and the area. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

SAS means side, angle, side. The angle must be the one enclosed by the two sides. If your known angle is opposite one of the sides instead, switch to the SSA mode.

Two sides and the included angle always fix exactly one triangle when the angle is between 0 and 180 degrees.

How to use SAS Triangle Calculator

  1. 1Pick the mode that matches what you know: SSS, SAS, ASA, AAS, SSA or Right triangle.
  2. 2Type the three values in the boxes. Sides a, b, c sit opposite angles A, B, C. In right mode fill exactly two boxes.
  3. 3Choose a length unit and whether angles are in degrees or radians. Changing the angle unit converts the numbers you already typed.
  4. 4Read the sides, angles, area and other measures, and compare them with the scaled diagram.
  5. 5Open the step-by-step working to see each law of cosines or law of sines line, then Heron’s formula for the area.
  6. 6In 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.

Third side and angles

The law of cosines gives the missing side, and then the angles follow.

c² = a² + b² − 2ab cos C
cos A = (b² + c² − a²) / (2bc)
B = 180° − A − C

Worked examples

  • a = 7, C = 50°, b = 9 → gives c = 7.0006

    Area 24.1304, from one half a b sin C.

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.

What is the included angle?

The angle formed where the two known sides meet. In this tool it is angle C, between sides a and b.

How do I find the area from SAS?

Multiply the two sides, multiply by the sine of the included angle and halve: area = ½ a b sin C.

More Triangle Calculator pages

  • Triangle CalculatorEnter any three valid values and get every side, every angle, the area and a drawing of the triangle.
  • SSS Triangle CalculatorEnter three sides and get all three angles, the area and a scaled drawing.
  • ASA Triangle CalculatorEnter two angles and the side between them to get the other two sides and the third angle.
  • AAS Triangle CalculatorEnter two angles and a non-included side to get every other side and angle.
  • SSA Triangle CalculatorEnter two sides and a non-included angle. The calculator shows every triangle that fits.
  • Right Triangle CalculatorEnter any two values of a right triangle and get the legs, the hypotenuse, both acute angles and the area.
  • Area of a Triangle CalculatorEnter three sides to get the area by Heron’s formula, with the heights and the steps.
  • Heron’s Formula CalculatorEnter the three sides and see the semi-perimeter, the product and the area worked out.
  • Law of Sines CalculatorEnter two angles and a side to find the others with the law of sines.
  • Law of Cosines CalculatorEnter two sides and the angle between them to find the third side and the other angles.
  • Isosceles Triangle CalculatorEnter the two equal sides and the base to get the angles, the height and the area.
  • Equilateral Triangle CalculatorEnter the side length to get the height, the area and both circle radii.

Ulkaa Apps

  • Ulkaa
  • Calendar
  • Task
  • Note
  • Disk
  • Call
  • Contact
  • Mail
  • Finance
  • Health
  • Office
  • Work
  • Workspace
  • Halum
  • Jaal
  • Islam
  • Game
  • Weather
  • Campus
  • News
  • Developer
  • Help
  • Account
About·Privacy·Terms·© 2026 Ulkaa

What is Ulkaa Tool?

Ulkaa Tool is a free collection of fast, privacy-friendly online tools from Ulkaa Network. Calculate your GPA and CGPA, edit rich text, count words and characters, replace text and more — all in your browser.

Every tool runs entirely in your browser — no sign-up, and your data never leaves your device.

Academic

  • GPA Calculator
  • CGPA Calculator
  • Citation Generator
  • Quiz Maker

Text

  • Rich Text Editor
  • Word Counter
  • Character Counter
  • Text Replacement

Ulkaa

  • Ulkaa
  • Ulkaa News
  • Ulkaa Campus
  • Ulkaa Tool
  • Ulkaa Game
  • Ulkaa Weather
  • Ulkaa Islam

Company

  • About Ulkaa
  • Careers
  • Support
  • Help
  • Privacy Policy
  • Terms of Service
  • Disclaimer
  • Sitemap
© 2015-26, Ulkaa. All rights reserved.
Facebook PageLinkedIn CompanyX (Twitter) ProfileInstagram ProfileGitHub Account