BrownMath.com → TI-83/84/89 → Solving Triangles
Updated 19 Nov 2021

# How to Solve Triangles on TI-83/84

Summary: Six numbers — three sides and three angles — determine a triangle. If you know any three of them, providing that at least one of the three is a side, you can find the other three. This program does just that, and as a bonus it finds the area of the triangle. (You can also get all three sides from two angles and the area.)

The program figures out whether you’re in degree or radian mode, and adjusts itself accordingly. The program doesn’t check for negative angles or two entered angles adding to more than 180° or 2π, so don’t be silly, m’kay?

The program works on all TI-83 Plus models, and all TI-84 models including the color models.

See also: For the computations, please see Solving Triangles. For the program code, see TRIANGLE.pdf in the accompanying TRIANGLE.zip file.

## Getting the Program

There are three methods to get the program into your calculator:

• If you have a TI-84, download TRIANGLE.zip (31 KB, updated 28 Dec 2016), and unzip it. Use the USB cable that came with your calculator, and the free TI Connect CE software from Texas Instruments, to transfer the TRIANGLE.8XP program to your calculator.
• If a classmate has the program on her calculator (any model TI-83+/84+), she can transfer it to yours, provided you both have a USB port or you both have a round I/O port. Connect the appropriate cable to both calculators, inserting each end firmly. On your calculator, press [`2nd` `x,T,θ,n` makes `LINK`] [`►`] [`ENTER`]. Then on hers press [`2nd` `x,T,θ,n` makes `LINK`] [`3`], select TRIANGLE, and finally press [`►`] [`ENTER`]. If you get a prompt about a duplicate program, choose Overwrite.
• Or, as a last resort, key in the program. See TRIANGLE.pdf and TRIANGLE_hints.htm in the TRIANGLE.zip file.

## Running the Program

It’s customary to refer to the angles as A, B, and C, and the sides as a, b, and c, such that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

When you run the TRIANGLE program, it prompts you to say which facts you know:

1. Angle-side-angle, two angles and the side between them.
2. Angle-angle-side, two angles and one of the sides not between them.
3. Side-angle-side, two sides and the angle between them.
4. Side-side-angle, two sides and one of the angles not between them. For some values, this case can have two solutions. The program will prompt you when that happens.
5. Side-side-side, all three sides known.
6. Angle-angle-angle-area, all three angles plus the area known.

Each case has either one unique solution, or none. The exception is SSA, which could have zero, one, or two solutions depending on the numbers. For about that, see Example 3 below, and Special Note: Side-Side-Angle in Trig without Tears.

### Example 1. Three Sides Known

Given the three sides of the triangle (menu item 5), you can use the program to find the three angles. Run the TRIANGLE program and select `5:SSS`. Enter the three sides, and the program gives you the area, the three angles, and the three sides. The first angle is opposite the first side, the second angle opposite the second side, and the third angle opposite the third side.

Angles are always displayed in degrees, with one decimal place on black-white screens, two decimal places on color screens. If one of the angles is ≥100°, the b&w display may be too narrow. In that case, just press [`ALPHA` `MATH` makes `A`], [`ALPHA` `APPS` makes `B`], or [`ALPHA` `PRGM` makes `C`] to display the angle. It’s the unrounded value, so you can also do this to get full precision. In the screen shot at right, angle C is shown.

In this case, of course you already know the sides. But in cases where the program is computing them, you might want more precision than the program displays. Press [`ALPHA` `x-1` makes `D`], [`ALPHA` `SIN` makes `E`], or [`ALPHA` `COS` makes `F`] for sides a, b, c, or [`ALPHA` `TAN` makes `G`] for the area.

### Example 2. Two Angles and Non-Included Side Known

When you know two angles and a side not between them, use `2:AAS`. Call the known side a; then the angle opposite it is A and the angle between them is B. Here a = 180, A = 31°, and B = 42°.

The third angle is 107°. The base of the triangle is 334.22; and the third side, opposite the 42° angle, is 233.85.

### Example 3. Two Sides and Non-Included Angle Known

Because this program helps you,
Because this program helps you,
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Suppose you know that two sides measure 8 and 10 units. You don’t know the angle between them, but you know that the angle opposite the 8-unit side is 45°. There are actually two triangles that meet these conditions, a larger triangle with B as an acute angle (<90°), and a smaller triangle with B as an obtuse angle (>90°). Those triangles are also shown separately here, in brown and blue respectively:

See Special Note: Side-Side-Angle for the exact conditions when SSA can give you zero, one, or two solutions.

Fortunately, the TRIANGLE program has these conditions programmed in for you: select `4:SSA`. When there are two possible solutions, the program prompts you to choose one. Here I’ve chosen the acute angle for B, which gives the larger triangle.

And here I’ve chosen the obtuse angle for B, which gives the smaller triangle.

### Example 4. Area and Two Angles Known

If you know any two angles, you know the third one and therefore you know the shape of the triangle. Then, any one side or the area will let you solve the triangle completely.

Example: Suppose you know that a triangle has angles of 30°, 50°, with area of 27.16. Make a sketch, using (spoiler alert!) 100° for the third angle and about 12 units for the long side.

Solution: Select `6:AA and Area`, and when prompted enter the two angles and the area. (If you know all three angles, it doesn’t matter which two you enter.)

Remember that angle A is opposite side a, so side a = 6 units is opposite angle A = 30°, side b = 9.19 units is opposite angle B = 50°, and side c = 11.82 units is opposite angle C = 100°.

## Program Variables

The TRIANGLE program uses several variables. They’re listed here because you might have occasion to use some of them after the program finishes:

• A, B, C are the entered and computed angles, in degrees.
• D, E, F are the entered and computed sides a, b, c.
• G is the area of the triangle, (ab/2) × sin C.
• H is the height of the triangle, b × sin A.
• R is the total angles of the triangle, either 180° or 2π radians.
• Z is 1 for color calculators and 0 for black-and-white calculators.

## What’s New?

• 19 Nov 2021: Updated the link to TI Connect CE software.
• 26 Sept 2016: Updated the “hints” file in the ZIP file, with no change to the actual program.
• 4 Apr 2016: Program v1.2:
• The program now works in degree or radian mode.
• Added case 6, three angles and area known, as suggested in email by Caroline McKnoe. There’s a worked-out example in this article.
• Overrode the calculator’s “Done” prompt with "Press Enter…” to stop the heading for the answers from scrolling off the top of the screen in Mathprint mode.
• 1 Jan 2016: Program v1.1:
• All TI-83+ and TI-84+ calculators are now supported. Lower case is now used for readability, and to distinguish sides a, b, c from angles A, B, C. There’s special formatting for color screens.