Almost every ACT triangle question runs on three facts: the angles inside a triangle add to 180 degrees, the area is one half the base times the height, and in a right triangle a squared plus b squared equals c squared. Layered on top are a few shortcuts that turn a multi-step problem into a five-second one. Geometry is 17 to 20 percent of the Math section, and triangles carry a big share of it.
Start with the angle sum
Whenever a figure hands you two angles, write the third one in immediately, even if the question hasn't asked for it yet. If a triangle has angles of 47 degrees and 68 degrees, the third is 180 minus 47 minus 68, which is 65 degrees. That habit solves some questions outright, and when it doesn't, the filled-in figure usually shows you what to do next.
Area, and why the height is often not a side
The area of a triangle is one half the base times the height, where the height is the perpendicular distance from the base up to the opposite vertex. Suppose a triangle has a base of 12, a slanted side labeled 10, and a dashed perpendicular segment labeled 8. The area is one half times 12 times 8, which is 48. Using the 10 would give 60, and that wrong answer will be sitting right there in the choices.
The Pythagorean theorem and two triples
In a right triangle, a squared plus b squared equals c squared, where c is the hypotenuse, the side across from the right angle. If the legs are 9 and 12, then 81 plus 144 is 225, and the square root of 225 is 15.
You can skip that arithmetic once you recognize the two triples the ACT loves: 3-4-5 and 5-12-13. The 9-12-15 triangle above is 3-4-5 with every side tripled, and a triangle with legs 10 and 24 is 5-12-13 doubled, so its hypotenuse is 26.
Special right triangles
Two right triangles have fixed side ratios, and memorizing them is non-negotiable since the ACT does not hand out a formula sheet.
- A 45-45-90 triangle has sides in the ratio 1 to 1 to the square root of 2. The two legs match, and the hypotenuse is a leg times the square root of 2. If the hypotenuse is 10, each leg is 10 divided by the square root of 2, which simplifies to 5 times the square root of 2, or roughly 7.07.
- A 30-60-90 triangle has sides in the ratio 1 to the square root of 3 to 2, in the order short leg, long leg, hypotenuse. The short leg sits across from the 30 degree angle. If the hypotenuse is 14, the short leg is 7 and the long leg is 7 times the square root of 3, or about 12.1.
Isosceles triangles and the exterior angle rule
An isosceles triangle has two equal sides, and the angles across from them are equal too, which works in reverse as well. If the vertex angle is 40 degrees, the remaining 140 degrees splits evenly, so each base angle is 70 degrees.
The exterior angle rule is the other quiet time-saver. Extend one side of a triangle and the angle formed outside equals the sum of the two interior angles it does not touch. If an exterior angle measures 115 degrees and one of those two remote interior angles is 40 degrees, the other is 75 degrees. The angle sum gets you there too, but this is one step instead of three.
Traps that cost points
- Assuming a triangle is right or isosceles because it looks that way. Without a right angle box, matching tick marks, or stated measures, you cannot use those properties.
- Treating a slanted side as the height. The height has to meet the base at a right angle, and it is often the faint dashed line.
- Mixing up which side is the hypotenuse. It is across from the right angle and is the longest side, so if your c comes out shorter than a leg, you set it up backwards.
A quick self-check
Try these three without a timer, then again with one. A triangle has angles of 32 degrees and 58 degrees; what is the third, and what does that tell you? A right triangle has one leg of 8 and a hypotenuse of 17; find the other leg. A 30-60-90 triangle has a short leg of 6; find the other two sides. The answers are 90 degrees, so it is a right triangle; 15, since 289 minus 64 is 225; and a long leg of 6 times the square root of 3 with a hypotenuse of 12.
If any of those felt slow, the fix is volume rather than theory. thirty-six groups triangle questions into concept sets so you can run them back to back instead of waiting for one to appear in a full section. From there, the natural next steps are similar triangles and right triangle trigonometry, which both build directly on what you just practiced. For how triangles fit into the rest of the section, see how much geometry is on the ACT.
Start practicing
Start with a free diagnostic, then drill your weak spots with 15-question quizzes and track how you're doing across Reading, English, and Math. Compare plans whenever you're ready to go further.
This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.