MathIsimple

Triangles Beginner

Master triangle fundamentals: definitions, classification by sides and angles, the angle sum theorem, triangle inequalities, and multiple area formulas—all in one comprehensive lesson.

Geometry
Beginner
~60 min
Triangle Calculator
Calculate area using base-height, Heron's formula, or SAS method

Use this embedded version to check base-height area calculations without leaving the lesson.

Calculator

The length of the triangle's base (any side can be the base)

cm

The perpendicular distance from the base to the opposite vertex

Quick Examples:

Results

Enter the base and height measurements, then click "Calculate Area" to find the triangle's area.

1. What is a Triangle?

A triangle is a polygon with exactly three vertices, three sides, and three interior angles. It is the simplest closed figure you can create with straight lines.

Standard Notation

  • Vertices: A, B, C (uppercase letters)
  • Sides: a, b, c (lowercase, opposite to corresponding vertex)
  • Angles: ∠A, ∠B, ∠C (at each vertex)
  • Triangle: △ABC

Key Convention

Side aa is opposite vertex A (connecting vertices B and C). This naming convention is used throughout geometry and trigonometry.

Why Triangles Are Rigid

If you fix the lengths of all three sides, the triangle's shape is completely determined. Unlike quadrilaterals (which can flex), triangles cannot deform without breaking a side. This is why triangles appear everywhere in construction: roof trusses, bridges, bicycle frames.

2. Classification by Sides and Angles

By Sides

Scalene

No equal sides, no equal angles

Isosceles

At least two equal sides, two equal base angles

Equilateral

All sides equal, all angles = 60°

By Angles

Acute

All angles < 90°

Right

Exactly one angle = 90°

Obtuse

One angle > 90°

Classification Tests Using Sides

For sides a, b, c with c being the longest:

  • a2+b2=c2a^2 + b^2 = c^2 → Right triangle
  • a2+b2>c2a^2 + b^2 > c^2 → Acute triangle
  • a2+b2<c2a^2 + b^2 < c^2 → Obtuse triangle
3. Angle Sum Theorem
∠A+∠B+∠C=180°\angle A + \angle B + \angle C = 180°

The sum of interior angles in any triangle is always exactly 180°.

Finding Missing Angles

If you know two angles, subtract their sum from 180°:

∠C=180°−∠A−∠B\angle C = 180° - \angle A - \angle B

Exterior Angle Theorem

An exterior angle equals the sum of two remote interior angles:

∠ext=∠remote1+∠remote2\angle_{ext} = \angle_{remote_1} + \angle_{remote_2}

Example: Algebraic Problem

If angles are x°, 2x°, and 3x°, find all angles:

x+2x+3x=180°x + 2x + 3x = 180°

6x=180°6x = 180°

x=30°x = 30°

Angles: 30°, 60°, 90° (a 30-60-90 right triangle!)

4. Triangle Inequality
a+b>candb+c>aanda+c>ba + b > c \quad\text{and}\quad b + c > a \quad\text{and}\quad a + c > b

The sum of any two sides must be greater than the third side.

Valid Triangle

Sides 3, 4, 5:
3 + 4 = 7 > 5 ✓
3 + 5 = 8 > 4 ✓
4 + 5 = 9 > 3 ✓

Invalid Triangle

Sides 2, 3, 7:
2 + 3 = 5 < 7 ✗
These lengths cannot form a triangle.

Quick Check

You only need to check that the sum of the two shorter sides is greater than the longest side. If that passes, all three inequalities are satisfied.

5. Triangle Area Formulas

Base-Height

S=12bhS = \frac{1}{2}bh

Any triangle; b is base, h is perpendicular height

Heron's Formula

S=s(s−a)(s−b)(s−c)S = \sqrt{s(s-a)(s-b)(s-c)}

Where s=a+b+c2s = \frac{a+b+c}{2} (semi-perimeter)

SAS Formula

S=12absin⁡CS = \frac{1}{2}ab\sin C

Two sides and included angle

Right Triangle

S=12abS = \frac{1}{2}ab

a and b are the two legs

Equilateral

S=34s2S = \frac{\sqrt{3}}{4}s^2

s is the side length

Isosceles

S=b4a2−b24S = \frac{b\sqrt{4a^2-b^2}}{4}

a = equal sides, b = base

Common Mistakes

  • • Using a slanted side instead of perpendicular height
  • • Forgetting to halve in base-height formula
  • • Using wrong angle (not the included angle) in SAS formula
  • • Mixing units (always convert to same unit first)

Practice Quiz

Practice Quiz
20
Questions
0
Correct
0%
Accuracy
1
What is the sum of interior angles in any triangle?
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2
If two angles of a triangle are 50°50° and 60°60°, what is the third angle?
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3
A triangle with all sides equal is called:
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4
Which triangle has exactly one angle of 90°90°?
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5
Can a triangle have sides of lengths 2, 3, and 7?
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6
The area of a triangle with base 10 and height 6 is:
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7
An isosceles triangle has base angles of 70°70° each. What is the vertex angle?
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8
Which formula calculates triangle area using only the three side lengths?
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9
A triangle with sides 3, 4, 5 is:
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10
What is the exterior angle at vertex A if the interior angle at A is 50°50°?
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11
In a triangle, the longest side is always opposite to:
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12
A triangle with all angles less than 90°90° is called:
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13
What is the area of an equilateral triangle with side length 4?
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14
If a triangle has angles xx, 2x2x, and 3x3x, what is xx?
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15
Which of these is NOT a valid triangle?
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16
The exterior angle of a triangle equals:
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17
A scalene triangle has:
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18
In a right triangle with legs 6 and 8, what is the area?
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19
Why are triangles used in construction and engineering?
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20
Can a triangle have two obtuse angles?
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