Mathematics
Grade 7
15 min
Open and closed shapes
Open and closed shapes
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1
Introduction & Learning Objectives
Learning Objectives
Define open and closed shapes.
Distinguish between open and closed shapes based on their endpoints.
Identify simple and non-simple closed shapes.
Classify various geometric figures as open or closed.
Understand the concepts of interior and exterior regions for closed shapes.
Draw examples of open, closed, simple closed, and non-simple closed shapes.
Have you ever noticed how some paths lead you back to where you started, while others leave you somewhere new? 🚶♀️🗺️ This idea is fundamental to understanding shapes!
In this lesson, you will learn to categorize shapes as either 'open' or 'closed' based on whether their starting and ending points meet. This basic geometric concept helps us understand boundaries, regions, and the properti...
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Key Concepts & Vocabulary
TermDefinitionExample
ShapeA form or outline of an object or a region, often defined by lines, curves, or points.A square, a circle, a wavy line.
Open ShapeA shape that does not start and end at the same point. It has distinct endpoints and does not enclose a region.A line segment, a spiral, the letter 'C'.
Closed ShapeA shape that starts and ends at the same point, forming a continuous path. It encloses a region and has no distinct endpoints.A circle, a square, a triangle, the letter 'O'.
Simple Closed ShapeA closed shape that does not intersect or cross itself at any point other than its starting/ending point.A triangle, a square, a circle.
Non-Simple Closed ShapeA closed shape that intersects or crosses itself at one or more points.A figure-eight, a star drawn with...
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Core Formulas
Rule for Closed Shapes
A path $P$ is classified as a closed shape if its starting point $S$ and its ending point $E$ are identical: $S = E$.
This rule helps identify shapes that form a complete loop, enclosing a region. If you can trace the shape without lifting your pen and end up exactly where you started, it's closed.
Rule for Open Shapes
A path $P$ is classified as an open shape if its starting point $S$ and its ending point $E$ are distinct: $S \neq E$.
This rule helps identify shapes that do not form a complete loop. If you trace the shape and end up at a different point from where you began, it's an open shape.
Rule for Simple Closed Shapes
A closed path $P$ is simple if it does not intersect itself at any point other than its start/end point.
This r...
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Challenging
A path is formed by the curve y = cos(x) from x=0 to x=2π, and then a straight line segment connecting the end of the curve back to the start. What is the classification of this complete path?
A.An open shape because it contains a curve.
B.non-simple closed shape because the curve goes up and down.
C.An open shape because it is made of two different parts.
D.simple closed shape.
Challenging
A pentagram is formed by connecting five points (A, B, C, D, E) in a circular order to form a star. The path is A→C→E→B→D→A. What is the most precise classification of this path?
A.simple open shape
B.simple closed shape
C.non-simple closed shape
D.collection of five open shapes
Challenging
If a non-simple closed shape, like a figure-eight, undergoes a dilation (is uniformly enlarged or shrunk), what happens to its classification?
A.It becomes a simple closed shape because the intersection point moves.
B.Its classification does not change; it remains a non-simple closed shape.
C.It becomes an open shape because the path is stretched.
D.Its classification depends on the scale factor of the dilation.
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