Two-Dimensional Figures - Seventh Grade
Polygons, Triangles, Quadrilaterals, Circles & Angles
1. Polygons - Definition and Classification
What is a Polygon?
A polygon is a CLOSED 2D shape made of
straight line segments
• Must be closed (no gaps)
• Made of straight lines only (no curves)
• At least 3 sides
Types of Polygons by Number of Sides
| Number of Sides | Name |
|---|---|
| 3 | Triangle |
| 4 | Quadrilateral |
| 5 | Pentagon |
| 6 | Hexagon |
| 7 | Heptagon |
| 8 | Octagon |
| 9 | Nonagon |
| 10 | Decagon |
Classification by Properties
Regular Polygon: All sides and angles are EQUAL
Example: Square, Equilateral triangle
Irregular Polygon: Sides or angles are DIFFERENT
Example: Rectangle, Scalene triangle
Convex Polygon: All interior angles < 180°
No vertices point inward
Concave Polygon: At least one angle > 180°
At least one vertex points inward
2. Classifying Triangles
Classification by SIDES
Equilateral Triangle: All 3 sides EQUAL
• All angles = 60°
Isosceles Triangle: 2 sides EQUAL
• 2 angles are equal
Scalene Triangle: All 3 sides DIFFERENT
• All 3 angles are different
Classification by ANGLES
Acute Triangle: All 3 angles < 90°
Right Triangle: One angle = 90°
Obtuse Triangle: One angle > 90°
Triangle Angle Sum
∠A + ∠B + ∠C = 180°
The sum of all interior angles in a triangle is ALWAYS 180°
3. Triangle Inequality Theorem
The Rule
a + b > c
The sum of any TWO sides must be
GREATER THAN the third side
For sides a, b, and c:
• a + b > c
• a + c > b
• b + c > a
Example
Can sides 3, 4, and 5 form a triangle?
Check: 3 + 4 = 7 > 5 ✓
Check: 3 + 5 = 8 > 4 ✓
Check: 4 + 5 = 9 > 3 ✓
YES, they can form a triangle!
Can sides 2, 3, and 8 form a triangle?
Check: 2 + 3 = 5, but 5 is NOT > 8 ✗
NO, they CANNOT form a triangle!
4. Classifying Quadrilaterals
What is a Quadrilateral?
A quadrilateral is a polygon with 4 sides
4 vertices and 4 angles
Sum of interior angles = 360°
Types of Quadrilaterals
1. Square
• All 4 sides EQUAL
• All 4 angles = 90°
• Opposite sides parallel
2. Rectangle
• Opposite sides EQUAL
• All 4 angles = 90°
• Opposite sides parallel
3. Parallelogram
• Opposite sides EQUAL and PARALLEL
• Opposite angles are equal
4. Rhombus
• All 4 sides EQUAL
• Opposite sides parallel
• Opposite angles equal
5. Trapezoid
• Exactly ONE pair of parallel sides
• Parallel sides called BASES
6. Kite
• Two pairs of adjacent sides equal
• No parallel sides
Quadrilateral Angle Sum
∠A + ∠B + ∠C + ∠D = 360°
Sum of interior angles = 360°
5. Interior Angles of Polygons
Sum of Interior Angles Formula
S = (n − 2) × 180°
Where:
S = Sum of interior angles
n = Number of sides
Each Interior Angle (Regular Polygon)
I = [(n − 2) × 180°] ÷ n
For regular polygons only (all angles equal)
Examples
| Polygon | Sides (n) | Sum Formula | Total |
|---|---|---|---|
| Triangle | 3 | (3−2)×180° | 180° |
| Quadrilateral | 4 | (4−2)×180° | 360° |
| Pentagon | 5 | (5−2)×180° | 540° |
| Hexagon | 6 | (6−2)×180° | 720° |
| Octagon | 8 | (8−2)×180° | 1080° |
6. Finding Missing Angles
In Triangles
Example 1: Two angles are 50° and 60°. Find the third angle.
Sum of angles = 180°
50° + 60° + x = 180°
110° + x = 180°
x = 70°
Third angle = 70°
Example 2 (Using Ratios): Angles are in ratio 2:3:4. Find each angle.
Let angles be 2x, 3x, and 4x
2x + 3x + 4x = 180°
9x = 180°
x = 20°
Angles: 2(20°) = 40°, 3(20°) = 60°, 4(20°) = 80°
Angles: 40°, 60°, 80°
In Quadrilaterals
Example: Three angles are 80°, 100°, and 90°. Find the fourth angle.
Sum of angles = 360°
80° + 100° + 90° + x = 360°
270° + x = 360°
x = 90°
Fourth angle = 90°
7. Parts of a Circle
Circle Vocabulary
Center
The point in the middle of the circle
Radius (r)
Distance from center to any point on the circle
Diameter (d)
Distance across circle through center
d = 2r (diameter = 2 × radius)
Chord
Line segment joining two points on the circle
Diameter is the longest chord
Arc
Part of the circle's circumference
Circumference (C)
Perimeter (distance around) the circle
C = 2πr or C = πd
Central Angle
A central angle is an angle whose VERTEX
is at the CENTER of the circle
• The two rays extend to the circle
• Measured in degrees
Sum of all central angles = 360°
Quick Reference: Angle Sums
| Shape | Sum of Interior Angles |
|---|---|
| Triangle | 180° |
| Quadrilateral | 360° |
| Pentagon | 540° |
| Hexagon | 720° |
| General (n sides) | (n−2) × 180° |
💡 Important Tips to Remember
✓ Polygon: Closed shape with straight sides
✓ Triangle angles: Always add to 180°
✓ Quadrilateral angles: Always add to 360°
✓ Triangle inequality: Sum of any 2 sides > third side
✓ Equilateral: All sides equal
✓ Isosceles: 2 sides equal
✓ Scalene: All sides different
✓ Regular polygon: All sides and angles equal
✓ Interior angles formula: (n−2) × 180°
✓ Circle parts: radius, diameter, chord, arc, circumference
✓ Central angles: Vertex at center, sum = 360°
🧠 Memory Tricks & Strategies
Triangle Angles:
"Three angles in a triangle make 180 - that's the single best angle fact!"
Quadrilateral Angles:
"Four sides, four angles too - together they make 360!"
Polygon Formula:
"Take the sides and minus two, times 180 - that's what to do!"
Triangle Types by Sides:
"Equilateral = Equal all, Isosceles = I see two, Scalene = Sides can't agree!"
Trapezoid:
"Trapezoid has one pair parallel - remember this and you'll excel!"
Diameter vs Radius:
"Diameter = 2 times radius - this relationship is very gracious!"
Master Two-Dimensional Figures! 🔺 ⬜ ⭕
Remember: (n−2) × 180° for interior angles!