Types of triangles by sides and angles

A triangle with sides of 5, 5 and 8 cm is isosceles and obtuse at the same time. The first name comes from its sides, the second from its largest angle, about 106°.

Two set squares, two different shapes

Look in a maths pencil case and you will often find a pair of plastic set squares. One has angles of 45°, 45° and 90°. The other has 30°, 60° and 90°. Both are right triangles, yet nobody would mix them up. The difference sits in the sides. The first one has two sides of equal length; on the second, all three sides are different. So a single triangle answers two separate questions. How do its three sides compare with each other? And how large is its largest angle? Each question gives the triangle a name, and the two names do not depend on each other.

First, what counts as a triangle. OpenStax puts it this way: join three points that do not lie on one straight line with line segments, and you have a triangle. The points are its vertices and the segments are its sides. A naming habit from the same book saves trouble later. Call the triangle ΔABC after its vertices, and give each side the lower-case letter of the vertex across from it, so side a lies opposite angle A, side b opposite angle B, side c opposite angle C. Unit 3 uses exactly this pairing when it shows that the longest side always faces the largest angle.

Sorting by sides gives three cases

If all three sides are equal, the triangle is equilateral. If two sides are equal, it is isosceles. In an isosceles triangle the third side is called the base, and the two angles at the ends of the base are the base angles. If no two sides are equal, the triangle is scalene. Drawings mark equal sides with matching tick marks rather than writing the lengths: one small dash on two sides says those two match, and a side with two dashes matches only other sides with two dashes.

Is an equilateral triangle also isosceles? Take the definition at its word, at least two equal sides, and the answer is yes: a triangle with three equal sides certainly has two. It is the same relation as between squares and rectangles. Every equilateral triangle is isosceles, but an isosceles triangle with sides 5, 5 and 8 is not equilateral. Some books define isosceles as exactly two equal sides; if you meet that version, the two names stop overlapping, so check which one your textbook uses.

Sorting by the largest angle gives three more

OpenStax sorts by angles in three short rules. In an acute triangle all three angles are less than 90°. A right triangle has one angle of exactly 90°, marked with a small square in the corner. An obtuse triangle has one angle between 90° and 180°. Why one, and not two? The three angles always add up to 180°, which the next unit proves. Two angles of 90° or more would already take the whole 180° and leave nothing for the third corner. So every triangle has at least two acute angles, and only the largest angle decides which of the three names it gets.

Which combinations can exist

Because the two names answer different questions, they combine. The table below shows one example for each pair. Two cells stay empty. An equilateral triangle has three equal angles, since equal sides face equal angles, so each angle is 180° ÷ 3 = 60°. It can never be right or obtuse. The 5, 5, 8 triangle from the top of the page lands in the isosceles and obtuse cell: its two base angles are about 36.9° each and the angle between the equal sides is about 106.3°.