What scenarios does the triangle explorer include?

Twelve scenarios across four categories: classification (equilateral, isosceles, acute, obtuse, right scalene), special right triangles (45-45-90, 30-60-90, 3-4-5, 5-12-13), trigonometric laws (law of sines, law of cosines), and a free-drag explore mode.

Can I reshape the triangle by dragging?

Yes. Tick the draggable checkbox in the top toolbar to enable vertex dragging for any scenario. The free-drag, law of sines, and law of cosines scenarios start draggable by default. The diagram, stats, and explanations all update in real time.

How do angle locks work?

Click the lock button next to an angle to freeze it at its current value. A number input and slider then let you set an exact degree measure. You can lock up to two angles; the third is computed automatically as 180 minus the sum of the other two.

What is the difference between law of sines and law of cosines?

Law of sines relates each side to the sine of its opposite angle and works best when you know two angles and a side. Law of cosines relates one side to the other two and the angle between them, useful when you know two sides and their included angle, or all three sides.

What do the colors in the diagram mean?

Red marks vertex A, side a (opposite A), and angle A. Amber marks vertex B, side b, and angle B. Blue marks vertex C, side c, and angle C. The same color scheme is used in the stats bar and the explanation formulas.

Triangle Explorer: Interactive Visualizer

Key Terms

• Vertex — a corner of the triangle. Labeled

A

,

B

,

C

.
• Side — segment between two vertices. Side

a

lies opposite vertex

A

, side

b

opposite

B

, and side

c

opposite

C

.
• Interior angle — the angle at a vertex, between the two sides meeting there. The three interior angles always sum to

180°

.
• Hypotenuse — in a right triangle, the side opposite the

90°

angle and the longest side.
• Pythagorean triple — three positive integers

(a, b, c)

satisfying

a^2 + b^2 = c^2

, like

3

-

4

-

5

or

5

-

12

-

13

.

Choosing a Scenario

The blue top bar groups twelve scenarios into four categories. Click any name to load that triangle:

• Classification — equilateral, isosceles, acute, obtuse, right scalene.
• Special — 45-45-90, 30-60-90, 3-4-5, 5-12-13.
• Trig laws — law of sines, law of cosines.
• Explore — free drag mode.

The active scenario stays highlighted. The diagram, intro text, and explanation panel on the right all refresh together. Click ↻ reset to return the current scenario to its starting shape.

Dragging Vertices

Many scenarios start static for clarity. Tick the draggable checkbox in the top bar to make all three vertices grabbable.

How dragging works:
• Hover a vertex — a circular halo confirms it is grabbable.
• Click and drag with mouse or touch.
• The triangle reshapes in real time, with sides, angles, and stats updating immediately.

The free-drag scenario starts with dragging enabled. Law of sines and law of cosines also start draggable, so you can confirm the laws hold for arbitrary shapes.

Locking Angles

The Lock angles controls let you fix one or two interior angles before dragging.

Per-angle controls:
• Click the lock button — the angle freezes at its current value.
• A number input appears with the locked value in degrees.
• A range slider lets you change the locked value continuously.

You can lock up to two angles at once. With two locked, the third is computed automatically as

180° - \angle_1 - \angle_2

. Locked angles show a dashed circle and a small lock icon at the vertex on the diagram.

Adjusting the Zoom

The Zoom controls in the second toolbar let you scale the diagram without changing the triangle:

• − decreases zoom by

10\%

down to

30\%

.
• + increases zoom by

10\%

up to

500\%

.
• fit snaps back to

100\%

when zoomed away.

Zooming affects only display size — the underlying coordinates, side lengths, and angle measures are unchanged. The grid in the background gives a stable visual reference at any zoom level.

Reading the Diagram

Each part of the SVG is color-coded so the same color always means the same thing.

Color mapping:
• Red — vertex

A

, side

a

, angle at

A

.
• Amber — vertex

B

, side

b

, angle at

B

.
• Blue — vertex

C

, side

c

, angle at

C

.

Visual cues:
• Curved arcs at each vertex show the interior angle, with the value in degrees.
• A small square mark replaces the arc when the angle is exactly

90°

.
• Side labels

a

,

b

,

c

sit on the outside of each segment.

Reading the Stats Bar

Below the diagram, a single line summarizes everything numeric:

• ∠A, ∠B, ∠C — interior angles in degrees, color-matched to vertices.
• a, b, c — side lengths in user units.
• P — perimeter, the sum of the three sides.
• Area — calculated from the vertex coordinates.

Drag any vertex (or change a locked angle) and watch the entire row update. This is the fastest way to verify properties like "the side opposite the largest angle is always the longest."

Reading the Explanation Panel

The right-hand panel changes contents per scenario, but the structure is consistent:

• Header — name of the active scenario.
• Intro — one-sentence summary of what is special about it.
• Explanation blocks — formula boxes and reasoning for the relevant theorems.

Examples by scenario:
• Equilateral — classification + angle sum.
• 3-4-5 — Pythagorean triple verification + trig ratios.
• Law of sines — the three ratios

a / \sin A

,

b / \sin B

,

c / \sin C

printed live.
• Law of cosines — both sides of

c^2 = a^2 + b^2 - 2ab\cos C

shown numerically.

Triangle Classifications

Triangles are classified two independent ways:

By angles:
• Acute — all three angles below

90°

.
• Right — one angle exactly

90°

.
• Obtuse — one angle above

90°

.

By sides:
• Equilateral — all three sides equal.
• Isosceles — exactly two sides equal.
• Scalene — all three sides different.

A triangle has one label from each list (e.g. right scalene, or the acute and isosceles pair). The explorer devotes a scenario to each: equilateral and obtuse complete the set.

For deeper coverage with proofs, see the triangle classifications page.

Pythagorean Theorem and Triples

In any right triangle with legs

a

,

b

and hypotenuse

c

:

a^2 + b^2 = c^2

A Pythagorean triple is a set of three positive integers satisfying this relation. The smallest examples:

• 3-4-5:

9 + 16 = 25

.
• 5-12-13:

25 + 144 = 169

.
• 8-15-17:

64 + 225 = 289

.

The explorer demonstrates the first two as dedicated scenarios, 3-4-5 and 5-12-13, with right scalene as the contrasting case where no triple exists. For full theory and proofs, see the Pythagorean theorem page.

Law of Sines and Law of Cosines

Two laws extend trigonometric reasoning beyond right triangles:

Law of sines — for any triangle:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Use it when you know two angles and one side, or two sides and a non-included angle. The law-of-sines scenario prints all three ratios live while you drag.

Law of cosines — generalizes Pythagoras:

c^2 = a^2 + b^2 - 2ab\cos C

Use it when you know two sides and the included angle, or all three sides. When

C = 90°

the cosine term vanishes and the formula reduces to Pythagoras — the law-of-cosines scenario lets you drag through that moment and watch the term disappear. For full coverage, see the law of sines page and the law of cosines page.

Related Concepts and Tools

Continue exploring with these connected resources:

• Trigonometric Functions —

\sin

,

\cos

,

\tan

and their reciprocals.
• Angle Explorer — visualize any angle, its quadrant, and its reference angle.
• Special Right Triangles — exact ratios for

45

-

45

-

90

and

30

-

60

-

90

.
• SOH CAH TOA — mnemonic for trig ratios in right triangles.
• Triangle Solver — compute missing sides and angles given partial information.
• Trigonometric Identities — Pythagorean, sum, and double-angle formulas built on these triangles.
• Pythagorean Identities —

\sin^2\theta + \cos^2\theta = 1

is the Pythagorean theorem on a hypotenuse of length

1

.
• Basic Trigonometric Identities — the reciprocal and quotient identities behind the ratios in the stats bar.

Scenario: Equilateral

The most symmetric triangle the explorer can show: three equal sides, three equal angles, and no way to tell one vertex from another.

Equilateral, frozen

Three equal sides force three equal angles — 60° each, the only split of 180° into three equal parts.

All three angles read

60°

, because the angle sum

180°

divides evenly three ways. That is the only value they can take — equilateral triangles are all the same shape, differing only in scale.

The scenario loads undraggable, since dragging any vertex would immediately destroy the property being illustrated. To deform it freely, switch to the free-drag scenario, which starts from a symmetric shape you are allowed to break.

Every equilateral triangle is also isosceles — the isosceles scenario shows the weaker condition, where only two sides match. Both sit under the side-based half of the classification scheme.

Scenario: Isosceles

Two sides equal, and with them the two angles opposite those sides — the base angles.

Isosceles, frozen

Base angles of 70° opposite the two equal sides, leaving 40° at the apex.

This frame is built with base angles of

70°

, which forces an apex of

40°

. The relationship runs both ways: equal sides imply equal base angles, and equal base angles imply equal sides.

The symmetry is what makes isosceles triangles useful in proofs. Dropping a perpendicular from the apex splits the figure into two congruent right triangles — the move behind several of the identity tools elsewhere in this section.

Compare with the equilateral scenario, where the condition is strengthened to all three sides, and with the acute scenario, where all three differ.

Scenario: Acute

All three angles below

90°

, and all three sides of different lengths — a scalene triangle that is also acute.

Acute scalene, frozen

Three arcs, none of them a square: every angle is under 90°, and all three sides differ.

The scenario exists to separate two classifications people often conflate. "Acute" describes the angles; "scalene" describes the sides. This triangle is both, and neither label implies the other.

Read the three arcs in the frame: each is drawn as a curve rather than the square mark that would signal a right angle. Compare with the obtuse scenario, where one arc opens past a straight edge, and with the right scalene scenario, where one arc becomes a square.

The full scheme, and why each triangle carries one label from each list, is in triangle classifications.

Scenario: Obtuse

One angle exceeds

90°

, which forces the other two to be small — their sum is whatever is left of

180°

.

Obtuse, frozen

One wide opening, and the longest side sitting directly opposite it.

A triangle can have at most one obtuse angle, since two would already exceed the total by themselves. The side opposite the obtuse angle is always the longest of the three, visible here as the side spanning the widest opening.

That "largest angle faces the longest side" rule is not specific to obtuse triangles — it holds everywhere, and the stats bar is the quickest way to confirm it while dragging.

The relationship is made exact by the law of cosines: when the angle passes

90°

, its cosine goes negative and the

-2ab\cos C

term starts adding to the opposite side rather than subtracting.

Scenario: Right Scalene

A right triangle whose three sides all differ — a right angle without any of the convenient ratios.

Right scalene, frozen

The square mark replaces an arc at the right angle. Legs 260 and 160 give an irrational hypotenuse.

The right angle is marked with a square instead of an arc, the explorer's signal that the angle is within

3°

of

90°

. The legs measure

260

and

160

in raw coordinates, so the hypotenuse is

\sqrt{260^2 + 160^2} \approx 305.3

— an irrational length, and the reason this scenario is not a triple.

It is the counterexample the special scenarios need. 3-4-5 and 5-12-13 are right triangles with whole-number sides; 45-45-90 and 30-60-90 have exact ratios. Most right triangles, this one included, have neither.

Pythagoras still applies, of course — see the theorem and its triples.

Scenario: 45-45-90

The isosceles right triangle: two equal legs, two

45°

angles, and a hypotenuse of exactly leg

\times \sqrt{2}

.

45-45-90, frozen

Equal legs, so the two acute angles split the remaining 90° evenly.

1 : 1 : \sqrt{2}

The ratio follows from Pythagoras in one line — with equal legs

x

, the hypotenuse is

\sqrt{x^2 + x^2} = x\sqrt{2}

. It is also half a square cut along its diagonal, which is the fastest way to remember it.

Being isosceles and right at once, this triangle belongs to two categories that the classification scheme treats separately. Its partner in the Special group is 30-60-90, the other exact-ratio right triangle.

Scenario: 30-60-90

Half of an equilateral triangle, cut down its axis of symmetry — which is exactly where its ratios come from.

30-60-90, frozen

The short leg faces the 30° angle and measures exactly half the hypotenuse.

1 : \sqrt{3} : 2

Slice an equilateral triangle in half and the

60°

angle survives, the cut halves one

60°

into

30°

, and the perpendicular creates the

90°

. The short leg is half the original side, hence the

1 : 2

relation with the hypotenuse; the long leg follows from Pythagoras.

These are the ratios behind the exact values of

\sin 30°

,

\cos 30°

and their companions — every exact trig value in the first quadrant comes from this triangle or from 45-45-90.

Scenario: 3-4-5

The smallest Pythagorean triple: three whole numbers that produce an exact right angle with no irrational lengths anywhere.

3-4-5, frozen

Drawn at 50× scale: legs of 200 and 150, hypotenuse 250 — the triple, enlarged.

3^2 + 4^2 = 9 + 16 = 25 = 5^2

The frame is drawn at

50

times that scale — legs of

200

and

150

coordinate units, hypotenuse

250

— because a triple stays a triple under scaling. Any multiple of

3

-

4

-

5

is another right triangle, which is why this one turns up in construction and surveying.

Its counterpart in the tool is 5-12-13, the next-smallest triple. Both are exact; right scalene shows the ordinary case where the hypotenuse is irrational. The general statement is in Pythagorean theorem and triples.

Scenario: 5-12-13

The second-smallest Pythagorean triple, and a much thinner triangle than the first.

5-12-13, frozen

At 20× scale. The legs differ by more than a factor of two, so the shape is visibly thin.

5^2 + 12^2 = 25 + 144 = 169 = 13^2

Here the frame is drawn at

20

times scale: legs of

100

and

240

, hypotenuse

260

. The shape is noticeably more elongated than 3-4-5 — the two legs differ by more than a factor of two, which pushes one acute angle down near

23°

.

Triples are rarer than they look. Of all right triangles, only a vanishingly small family has three integer sides; the rest look like right scalene.

Scenario: Law of Sines

A draggable triangle carrying three ratios that stay equal no matter how you reshape it.

Law of sines, frozen at its opening shape

A deliberately irregular triangle: the three ratios stay equal even here, and stay equal as you drag.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

The scenario starts draggable on purpose. The panel prints all three ratios live, so dragging a vertex is a running experiment: the individual sides and angles change constantly while the three quotients stay locked together.

The common value is not arbitrary — it equals the diameter of the triangle's circumscribed circle. Use the law when you know two angles and a side, or two sides and a non-included angle.

Its companion is the law of cosines, which handles the cases this one cannot. Both are stated in full under law of sines and law of cosines.

Scenario: Law of Cosines

The generalization of Pythagoras to triangles with no right angle, again draggable so both sides of the equation can be watched at once.

Law of cosines, frozen at its opening shape

No right angle anywhere, which is precisely when the −2ab·cos C correction term does work.

c^2 = a^2 + b^2 - 2ab\cos C

The correction term

-2ab\cos C

is the whole story. At

C = 90°

the cosine is zero, the term vanishes, and the formula collapses into Pythagoras. Below

90°

the cosine is positive and the term shortens

c

; above

90°

it goes negative and lengthens

c

— which is why obtuse triangles have such a long side opposite the wide angle.

Drag the vertices and watch the two computed sides of the equation stay equal. Use the law when you know two sides and the included angle, or all three sides and want an angle. The law of sines covers the complementary cases.

Scenario: Free Drag

No constraint at all: three vertices you can move anywhere, with every measurement recomputed as you go.

Free drag, frozen at its starting shape

A symmetric opening position with nothing pinned — every vertex is grabbable from the first click.

The panel here runs every explanation block at once — classification, angle sum, trig ratios and Pythagoras — so the labels update live as the shape crosses between categories. Drag a vertex until one angle passes

90°

and watch the classification flip from acute to right to obtuse.

This is also where angle locking earns its keep. Fix one or two angles and the drag becomes constrained: the triangle can still change size and orientation, but not the angles you pinned.

It is the scenario to reach for after the fixed ones, once the question stops being "what does an isosceles triangle look like" and becomes "what stays true for every triangle".