→ TI-83/84/89 → Solving Triangles
Updated 19 Nov 2021 (What’s New?)

How to Solve Triangles on TI-83/84

Copyright © 2015–2024 by Stan Brown,

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 file.


Getting the Program

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

Running the Program

splash screen for TRIANGLE program
selection of the three known items; see text

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

triangle with sides 238, 180, and 340 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.

input screen showing sides 238, 180, and 340       output screen showing area 20365.17; angles 41.7, 30.2, 108; and the three sides

continuation, with display showing angle C equal to 108.0558544 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°.

triangle with angle A = 31°, angle B = 42°, and side a (opposite angle A) = 180     input screen showing angle A = 31, angle  = 42, side a = 180       output screen showing area 20127.18; angles 31, 42, 107; and sides 180, 233.85, 334.22

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

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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:

triangle with known sides and angles marked, showing two possible configurations       possible solution with acute angle B, making a larger triangle       possible solution with obtuse angle B, making a smaller triangle

See Special Note: input screen, showing entry of two sides ad non-included angle 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.

menu screen, choosing acute angle for B       output screen, showing area = 38.23, angles = 45, 62.1, 72.9°; sides = 8, 10, 10.81       possible solution with acute angle B, making a larger triangle

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

menu screen, choosing obtuse angle for B       output screen, showing area = 11.77, angles = 45, 117.9, 17.1°; sides = 8, 10, 3.33       possible solution with obtuse angle B, making a 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.)

input screen, showing angles 30° and 50°, area 27.16        output screen, showing area 27.16, angles 30°, 100°, 50°, and sides 6, 11.82, 9.19

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:

What’s New?

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