Interactive Triangle Explorer | Visual Geometry Learning

Understanding Triangles

Interactive lesson with clear examples, activities, and instant checks.

Lesson 1 of 7
Lesson 1

Triangle Types

A triangle can be named by its side lengths and by its angles.

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Try it: Drag the orange points. Watch the side lengths, angles, and triangle names update.
Drag any orange point
Live Measurements

Sides

AB—
AC—
BC—

Angles

∠A—
∠B—
∠C—
A + B + C180°

Triangle Type

By sides—
By angles—
By sides
  • Equilateral: 3 equal sides.
  • Isosceles: 2 equal sides.
  • Scalene: no equal sides.
By angles
  • Acute: all angles are less than 90°.
  • Right: one angle is exactly 90°.
  • Obtuse: one angle is greater than 90°.
Engineers use triangles in bridges, roofs, towers, and bicycle frames because triangles are strong and keep their shape.
The type of a triangle depends on its sides and angles. Change one vertex and the whole triangle can change.
Which triangle has three equal sides?
Lesson 2

Angle Sum Rule

The three interior angles of every triangle always add up to 180°.

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Try it: Drag the triangle. The angle values change, but the total remains 180°.
Angle Total

Angles

∠A—
∠B—
∠C—

Total

A + B + C180°

Use it

Missing angle180° − known angles
If two angles are 50° and 70°, the third angle is 180° − 50° − 70° = 60°.
A triangle has angles 40° and 80°. What is the third angle?
Lesson 3

Exterior Angle Rule

An exterior angle equals the sum of the two opposite interior angles.

Exterior Angle Comparison

Opposite angles

∠A55°
∠B75°

Add them

A + B130°

Exterior angle

Outside angle130°
Exterior angle = the two remote interior angles added together. Example: 55° + 75° = 130°.
The two opposite angles are 45° and 65°. What is the exterior angle?
Lesson 4

Triangle Inequality

A triangle only works when the two shorter sides can reach each other above the longest side.

Move the side lengths
a = BC
b = AC
c = AB
Side a
6
Side b
7
Side c
8
✓
Shortest + middle side = ———
☝
Try it: Change the three side lengths. The page will sort them, test them, and tell you if they can close to make a triangle.
1
Sort the sides.Put the three side lengths in order from shortest to longest.
2
Add the two shorter sides.These two sides must be long enough to reach each other.
3
Compare with the longest side.If shortest + middle > longest, the sides can close and form a triangle.
Why this works

Good example

5 + 6 > 7Triangle

Bad example

2 + 3 < 8No triangle

Remember

Two short sides must be greater than the longest side.Rule
Which sides can form a triangle?
Lesson 5

Pythagorean Theorem

In a right triangle, the squares of the two shorter sides add up to the square of the longest side.

Right Triangle Calculation

Choose legs

a6 cm
b8 cm

Squares

a² + b²100
c²100

Answer

c10 cm
a² + b² = c². The side c is the hypotenuse: the longest side opposite the right angle.
A right triangle has legs 3 cm and 4 cm. What is the hypotenuse?
Lesson 6

Right Triangle: Side Names

In a right triangle, the sides do not only have letter names like AB, AC, and BC. They also have useful role names: hypotenuse, adjacent, and opposite.

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Try it: Choose the reference angle. The hypotenuse stays the same, but adjacent and opposite can change.
Side Roles

Hypotenuse

AClongest side
Positionopposite 90°

Adjacent

BCnext to angle
Used withcos

Opposite

ABacross from angle
Used withsin

Hypotenuse

The longest side in a right triangle. It is always opposite the 90° angle.

Pythagoras: a² + b² = c²

Adjacent

The side next to the reference angle. It touches the angle, but it is not the hypotenuse.

cos θ = adjacent / hypotenuse

Opposite

The side across from the reference angle. It does not touch the reference angle.

sin θ = opposite / hypotenuse
Simple method: First find the right angle. The side opposite it is the hypotenuse. Then choose your reference angle. The side touching that angle is adjacent. The side across from it is opposite.
In a right triangle, where is the hypotenuse?
Lesson 7

Final Challenge

Choose an exam, answer each question, and use the lesson reference button whenever you need to review the idea.

Triangle Master Exams
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Each exam has 15 real, unique questions covering the full triangle lesson.