Triangles

The triangle is the simplest shape you can build from straight lines — you need at least three sides to enclose any area at all — and yet it quietly holds up almost every bridge, roof, and radio tower you have ever seen. That combination of simplicity and strength is not a coincidence. Because a triangle's three sides completely lock its three angles into place, it is the only polygon that cannot be flexed out of shape without breaking or bending a side. This single property, called rigidity, is why engineers reach for triangles first, and why so much of geometry, trigonometry, and even the coordinate methods you will meet later are built on triangular foundations.

In this page we will treat triangles the way a working mathematician does: not as a list of facts to memorize, but as a small, tightly connected theory. We will see why the angles of any triangle sum to a straight angle, why two triangles that agree on the right pieces of information must be identical, and why the most famous relationship in mathematics — the Pythagorean theorem — is true, not just that it is.

Learning Objectives

By the end of this page, you should be able to:

• Classify triangles by their sides (scalene, isosceles, equilateral) and by their angles (acute, right, obtuse).
• Explain and use the fact that the interior angles of any triangle sum to $180°$.
• Decide when two triangles are congruent using SSS, SAS, ASA (and AAS), and recognize why SSA and AAA fail.
• Recognize similar triangles and use the ratio of corresponding sides to find unknown lengths.
• State the Pythagorean theorem, follow a proof-sketch of why it holds, and apply it (and its converse).
• Compute triangle area with the base-height, Heron, and trigonometric formulas, and choose the right one for the data you have.

Quick Answer

A triangle is a closed figure with three straight sides and three interior angles that always add to $180°$. Triangles are named by their sides — scalene (all different), isosceles (two equal), equilateral (all equal) — or by their largest angle — acute, right, or obtuse. Two triangles are congruent (identical in size and shape) if they match on any of SSS, SAS, ASA, or AAS; they are similar (same shape, possibly different size) if their angles match, which forces their sides into equal ratios. For a right triangle with legs $a$, $b$ and hypotenuse $c$, the Pythagorean theorem states $a^2 + b^2 = c^2$. Area is most often computed as $\tfrac{1}{2}\,\text{base}\times\text{height}$, but Heron's formula and $\tfrac{1}{2}ab\sin C$ let you find area from other combinations of information.

Where It Came From

Triangles were studied for a very practical reason: measuring the land and building on it. The word "geometry" literally means "earth-measurement," and its earliest driver was the need to re-survey farmland in ancient Egypt after the Nile's annual floods erased boundary markers. Right angles and triangular relationships let surveyors reconstruct fields reliably, and the famous "rope-stretchers" are thought to have used a knotted loop divided into segments of 3, 4, and 5 units to snap a perfect right angle in the ground.

That 3-4-5 fact points to something remarkable: the relationship we call the Pythagorean theorem was known and used more than a thousand years before Pythagoras. A Babylonian clay tablet called Plimpton 322 (roughly 1800 BCE) lists what are clearly Pythagorean triples — whole-number side lengths satisfying $a^2+b^2=c^2$ — showing the Babylonians had a systematic grip on the relationship. In India, the Sulba Sutras (texts on altar construction, composed several centuries BCE and rooted in older tradition) state the theorem explicitly to build altars of prescribed area, and Chinese mathematicians recorded it as the gougu rule. Pythagoras and his school (around 500 BCE) are traditionally credited not with discovering the relationship but with an early proof — a shift from "this works" to "this must be true," which is the birth of deductive mathematics. A few centuries later, Euclid's Elements (around 300 BCE) organized all of this into the axioms-and-proofs system that still shapes how geometry is taught today.

The deeper motivation behind studying triangles, though, is structural. Any shape with four or more sides can be deformed while keeping its side lengths fixed — push on a square and it collapses into a rhombus. A triangle cannot: fix the three lengths and the shape is determined completely (this is exactly why SSS congruence works). That rigidity is the reason triangles became the fundamental building block of trusses, frames, and the entire edifice of trigonometry.

Types of Triangles and the Angle Sum

Classifying by sides. A scalene triangle has three different side lengths; an isosceles triangle has (at least) two equal sides; an equilateral triangle has all three equal. A neat companion fact: equal sides sit opposite equal angles. So an equilateral triangle also has three equal angles — each $60°$ — and an isosceles triangle has two equal "base angles."

Classifying by angles. An acute triangle has all three angles less than $90°$; a right triangle has exactly one $90°$ angle; an obtuse triangle has one angle greater than $90°$. A triangle can have at most one right or obtuse angle — you will see why in a moment.

The angle sum. In any triangle the three interior angles add to $180°$. Here is why, using parallel lines. Take triangle $ABC$ and draw a line through vertex $A$ parallel to the opposite side $BC$.

$$\text{angle at } B + \text{angle at } A + \text{angle at } C = 180°$$

The line through $A$ creates two "alternate interior angles" with the transversals $AB$ and $AC$: the angle on one side of $A$ equals the angle at $B$, and the angle on the other side equals the angle at $C$. Those two angles plus the actual angle at $A$ lie along a straight line through $A$, so together they make a straight angle, $180°$. Since they equal $B + A + C$, the three interior angles sum to $180°$. This is also why only one angle can be $90°$ or more: two such angles would already reach or exceed $180°$, leaving nothing for the third.

Worked example. In a triangle, one angle is $x$, the second is $2x$, and the third is $x + 30°$. Find all three.

$$x + 2x + (x + 30°) = 180° \;\Rightarrow\; 4x + 30° = 180° \;\Rightarrow\; 4x = 150° \;\Rightarrow\; x = 37.5°$$

So the angles are $37.5°$, $75°$, and $67.5°$. Check: $37.5 + 75 + 67.5 = 180°$. Because all three are less than $90°$, this is an acute triangle.

Congruence: When Are Two Triangles Identical?

Two triangles are congruent if one can be laid exactly on top of the other — same side lengths and same angles. You do not need to check all six pieces (three sides, three angles). Certain minimal combinations are enough to force congruence:

• SSS (Side-Side-Side): all three sides match.
• SAS (Side-Angle-Side): two sides and the angle between them match.
• ASA (Angle-Side-Angle): two angles and the side between them match.
• AAS (Angle-Angle-Side): two angles and a non-included side match (this works because the third angle is forced by the $180°$ rule).

Two combinations do not guarantee congruence. AAA (all angles equal) only guarantees the same shape, not the same size — that is similarity, not congruence. And SSA (two sides and a non-included angle) is ambiguous: there can be two different triangles fitting the data, which is the famous "ambiguous case."

Worked example. Triangles $ABC$ and $DEF$ have $AB = DE = 6$, angle $A =$ angle $D = 50°$, and $AC = DF = 8$. Are they congruent? The equal angle ($50°$) is between the two equal sides in each triangle, so this is SAS. Yes — the triangles are congruent, and therefore $BC = EF$ as well, even though we were never told those lengths. That is the power of a congruence rule: match the right three pieces and everything else follows automatically.

Similarity: Same Shape, Different Size

Two triangles are similar when their angles are equal, one to one. Similar triangles are scaled copies: corresponding sides are all in the same ratio $k$, called the scale factor. Because the angles alone fix the shape, checking two angles is enough (the third is forced) — this is the AA similarity criterion.

If triangle $ABC$ is similar to triangle $DEF$ with scale factor $k$, then

$$\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC} = k, \qquad \text{and the areas scale as } k^2.$$

Worked example. A vertical stick $1.5\text{ m}$ tall casts a shadow $2\text{ m}$ long. At the same moment, a flagpole casts a shadow $18\text{ m}$ long. How tall is the flagpole?

The Sun's rays hit both objects at the same angle, and both stand vertically, so the two right triangles (object + shadow + ray) are similar by AA. Corresponding sides are in proportion:

$$\frac{\text{pole height}}{\text{pole shadow}} = \frac{\text{stick height}}{\text{stick shadow}} \;\Rightarrow\; \frac{h}{18} = \frac{1.5}{2} \;\Rightarrow\; h = 18 \times 0.75 = 13.5\text{ m}.$$

This "shadow method" is exactly how ancient astronomers estimated the heights of pyramids and, later, distances that could never be measured directly.

The Pythagorean Theorem

For a right triangle with legs of length $a$ and $b$ and hypotenuse (the side opposite the right angle) of length $c$:

$$a^2 + b^2 = c^2.$$

A proof-sketch by rearrangement. Take four identical copies of the right triangle and arrange them inside a big square whose side is $a+b$, with the right-angle corners pointing outward so the four hypotenuses enclose a tilted square in the middle. The tilted inner square has side $c$, so its area is $c^2$. Now compute the big square's area two ways.

• Whole square: side $a+b$, so area $(a+b)^2 = a^2 + 2ab + b^2$.
• The pieces: four triangles, each of area $\tfrac{1}{2}ab$, plus the inner square $c^2$. That totals $4\cdot\tfrac{1}{2}ab + c^2 = 2ab + c^2$.

Setting the two expressions equal:

$$a^2 + 2ab + b^2 = 2ab + c^2 \;\Rightarrow\; a^2 + b^2 = c^2.$$

The cross-terms $2ab$ cancel, leaving exactly the theorem. Notice we did not measure anything — the result is forced by area accounting, which is what makes it a proof.

Worked example (finding the hypotenuse). A right triangle has legs $a = 5$ and $b = 12$. Then

$$c^2 = 5^2 + 12^2 = 25 + 144 = 169 \;\Rightarrow\; c = \sqrt{169} = 13.$$

Worked example (finding a leg). A $10\text{ m}$ ladder leans against a wall with its foot $6\text{ m}$ from the base. How high up the wall does it reach? Here the ladder is the hypotenuse:

$$h^2 + 6^2 = 10^2 \;\Rightarrow\; h^2 = 100 - 36 = 64 \;\Rightarrow\; h = 8\text{ m}.$$

The converse is just as useful: if $a^2 + b^2 = c^2$ for the three sides of a triangle, then the triangle must have a right angle opposite $c$. This is exactly the rope-stretchers' trick — $3^2 + 4^2 = 9 + 16 = 25 = 5^2$, so a 3-4-5 triangle is guaranteed to contain a perfect right angle.

Area of a Triangle

The workhorse formula uses a base and the perpendicular height to that base:

$$\text{Area} = \tfrac{1}{2}\,b\,h.$$

Any triangle is exactly half of a parallelogram (or rectangle) with the same base and height, which is where the $\tfrac{1}{2}$ comes from. When you do not know the height, two other formulas help:

• Heron's formula, using the three sides. With semi-perimeter $s = \tfrac{a+b+c}{2}$,

$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}.$$

• The trigonometric formula, using two sides and the angle between them:

$$\text{Area} = \tfrac{1}{2}\,a\,b\,\sin C.$$

Worked example (base-height). A triangle has base $10\text{ cm}$ and height $6\text{ cm}$: Area $= \tfrac{1}{2}\times 10 \times 6 = 30\text{ cm}^2$.

Worked example (Heron). Find the area of a triangle with sides $7$, $8$, and $9$. First $s = \tfrac{7+8+9}{2} = 12$. Then

$$\text{Area} = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12\cdot 5\cdot 4\cdot 3} = \sqrt{720} \approx 26.83.$$

Real-World Applications

• Structural engineering. Roof trusses, cranes, transmission towers, and geodesic domes are built from triangles precisely because triangles cannot be deformed without changing a side length. A rectangular frame racks and collapses; add a diagonal to split it into two triangles and it becomes rigid.
• Navigation and surveying (triangulation). Measure one baseline and two angles, and similar-triangle and trigonometric relationships pin down a distant point without walking to it. GPS uses a 3-D version of the same idea.
• Computer graphics. Every 3-D model on a screen — games, films, CAD — is a mesh of tiny triangles, because a triangle is always flat and always convex, making it the ideal unit for fast rendering.
• Astronomy. The stick-and-shadow (similar triangles) method and parallax (using the triangle formed by two viewpoints and a star) let us estimate sizes and distances far beyond direct measurement.
• Everyday problem-solving. Checking whether a doorway is "square," figuring out whether a ladder is safely angled, or cutting a diagonal brace for a gate all use the Pythagorean theorem and the rigidity of triangles.

Common Mistakes

1. Applying $a^2 + b^2 = c^2$ to non-right triangles. The theorem holds only when there is a $90°$ angle, and $c$ must be the side opposite it. For other triangles you need the Law of Cosines. Correction: confirm the right angle first, and always place the hypotenuse (the longest side) as $c$.

2. Thinking equal angles means equal triangles. AAA gives you the same shape but not the same size — two equilateral triangles of different sizes have identical angles. Correction: matching angles proves similarity; you need at least one matching side (as in ASA, AAS, or SAS) for congruence.

3. Trusting SSA. Students often assume two sides and any angle fix a triangle. When the known angle is not between the two sides, two different triangles can fit the data. Correction: only SSS, SAS, ASA, and AAS guarantee congruence; treat SSA as ambiguous.

4. Using a slanted side as the "height" in the area formula. The $h$ in $\tfrac{1}{2}bh$ is the perpendicular distance from the base to the opposite vertex, not the length of another side. Correction: if you only have side lengths, use Heron's formula; if you have two sides and the included angle, use $\tfrac{1}{2}ab\sin C$.

Comparison and Connections

Idea	What it guarantees	Key requirement
Congruence (SSS, SAS, ASA, AAS)	Identical size and shape	A matching side somewhere
Similarity (AA)	Same shape, sides in a fixed ratio	Matching angles
Pythagorean theorem	Relates the three sides	A right angle
Law of Cosines (later)	Relates sides and any angle	Works for all triangles

The Pythagorean theorem is actually a special case of the Law of Cosines, $c^2 = a^2 + b^2 - 2ab\cos C$: when $C = 90°$, $\cos C = 0$ and the last term vanishes, recovering $c^2 = a^2 + b^2$. Similarity, meanwhile, is the doorway to trigonometry — the sine, cosine, and tangent ratios are well-defined only because all right triangles sharing an angle are similar, so the side ratios depend on the angle alone.

Practice Questions

Recall

State the three congruence rules that involve at least one side and one angle, and give the Pythagorean theorem. Answer: SAS, ASA, AAS (also pure-side SSS). Pythagorean theorem: $a^2 + b^2 = c^2$ for legs $a$, $b$ and hypotenuse $c$ in a right triangle.

Understanding

Explain why the angles of a triangle cannot include two obtuse angles. Answer: Each obtuse angle exceeds $90°$, so two of them already exceed $180°$. Since all three angles must total exactly $180°$, at most one can be obtuse (the same reasoning limits a triangle to one right angle).

Application

A right triangle has one leg $9$ and hypotenuse $15$. Find the other leg and the area. Answer: $b^2 = 15^2 - 9^2 = 225 - 81 = 144$, so $b = 12$. The legs are perpendicular, so area $= \tfrac{1}{2}\times 9 \times 12 = 54$.

Analysis

A person $1.8\text{ m}$ tall stands so that the tip of their shadow just meets the tip of a tree's shadow. The person is $3\text{ m}$ from the meeting point, and the tree is $12\text{ m}$ from it. How tall is the tree, and which principle did you use? Answer: The person-triangle and tree-triangle share the Sun's ray angle and both are vertical, so they are similar (AA). Then $\tfrac{h}{12} = \tfrac{1.8}{3} = 0.6$, giving $h = 12 \times 0.6 = 7.2\text{ m}$.

FAQ

Is every equilateral triangle also isosceles? Yes. "Isosceles" means at least two equal sides, and an equilateral triangle has three, so it satisfies the condition. It is a special case, just as a square is a special rectangle.

Why do the angles always add to exactly $180°$ — could a triangle ever be different? On a flat (Euclidean) surface, no — the parallel-line argument nails it down. But on a curved surface the sum changes: a triangle drawn on a sphere has angles summing to more than $180°$. Ordinary school geometry assumes flatness, so $180°$ holds there.

How is congruence different from similarity in plain terms? Congruent triangles are exact copies — same size and shape. Similar triangles are photocopies that may be enlarged or shrunk — same shape, proportional sizes. Every pair of congruent triangles is also similar (with scale factor $1$), but not the other way around.

When should I use Heron's formula instead of $\tfrac{1}{2}bh$? Use $\tfrac{1}{2}bh$ when you know a base and its perpendicular height. Use Heron's formula when you only know the three side lengths and no height. If you know two sides and the angle between them, $\tfrac{1}{2}ab\sin C$ is quickest.

Does the Pythagorean theorem work for triangles that are not right-angled? No. It is exclusively for right triangles. For any other triangle use the Law of Cosines, of which the Pythagorean theorem is the right-angle special case.

Why are triangles used in construction instead of squares? Because a triangle is rigid: its three side lengths completely determine its shape, so it cannot flex. A four-sided frame can be pushed out of square while its sides stay the same length, so builders add diagonal braces to break rectangles into triangles.

Quick Revision

• Interior angles of any (flat) triangle sum to $180°$; equal sides sit opposite equal angles.
• Side types: scalene / isosceles / equilateral. Angle types: acute / right / obtuse (at most one right or obtuse).
• Congruence (identical): SSS, SAS, ASA, AAS. Not SSA, not AAA.
• Similarity (same shape): AA; corresponding sides share a ratio $k$; areas scale as $k^2$.
• Pythagoras: $a^2 + b^2 = c^2$ for right triangles; converse identifies right angles (e.g. 3-4-5).
• Area: $\tfrac{1}{2}bh$; Heron $\sqrt{s(s-a)(s-b)(s-c)}$ with $s=\tfrac{a+b+c}{2}$; or $\tfrac{1}{2}ab\sin C$.
• Triangles are the rigid building block of structures — three sides fix the shape completely.

Related Topics

Prerequisites

• Lines and Angles
• Variables and Expressions

Related Topics

• Right-Triangle Trigonometry
• Coordinate Geometry
• Area and Volume

Next Topics

• Circles
• The Unit Circle