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Geometry - Open and Closed Figures

Grade 3ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An Open Figure is a shape that does not end at the same point where it started. There is a gap between the starting and ending points, meaning you can get to the inside from the outside without crossing any lines.

An open figure showing a polyline where the start and end points do not meet.
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A Closed Figure is a shape that starts and ends at the exact same point. It completely encloses an area, creating an 'inside' and an 'outside' separated by the boundary.

A closed diamond shape representing a closed figure.
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Figures can be made of straight lines (like a triangle), curved lines (like a circle), or a combination of both. As long as the path is continuous and returns to the start, it is closed.

A circle as an example of a closed figure made of a curved line.
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A Simple Closed Figure is a closed shape that does not cross itself. Polygons are examples of simple closed figures made entirely of line segments.

A square representing a simple closed figure made of four line segments.

📐Formulae

Closed Figure Condition: Start Point=End Point\text{Closed Figure Condition: Start Point} = \text{End Point}

Open Figure Condition: Start Point≠End Point\text{Open Figure Condition: Start Point} \neq \text{End Point}

Triangle sides=3\text{Triangle sides} = 3

Quadrilateral sides=4\text{Quadrilateral sides} = 4

Pentagon sides=5\text{Pentagon sides} = 5

Hexagon sides=6\text{Hexagon sides} = 6

💡Examples

Problem 1:

Rahul draws a shape starting at point PP. He draws a straight line to point QQ, then to point RR, and finally stops at point RR. Is this an open or a closed figure?

Solution:

Step 1: Identify the starting point, which is PP. Step 2: Identify the ending point, which is RR. Step 3: Compare the points. Since point PP is not the same as point RR (P≠RP \neq R), there is an opening between them.

Explanation:

Because the figure does not return to its starting point, it is classified as an open figure.

Problem 2:

A polygon has 44 vertices (corners) and all its sides are straight lines of equal length. If the figure is closed and simple, name the figure and identify how many line segments it has.

Solution:

Step 1: A closed figure with 44 straight sides is called a quadrilateral. Step 2: Since all 44 sides are of equal length and it is a simple closed figure, it is specifically a square. Step 3: Count the line segments: Side 1+Side 2+Side 3+Side 4=4\text{Side 1} + \text{Side 2} + \text{Side 3} + \text{Side 4} = 4 segments.

Explanation:

A square is a simple closed figure (polygon) made of 44 equal line segments where the start and end points meet.

Problem 3:

Look at the figure provided. It consists of three line segments forming a shape. Does this figure represent an open or closed figure? Explain why.

A U-shaped polyline that is not joined at the ends.

Solution:

The figure is an Open Figure.

Explanation:

By observing the path, the starting point is at the top left and the path moves to the right, then down, then left. However, it stops before reaching the starting point. Since Start Point≠End Point\text{Start Point} \neq \text{End Point}, it remains open.

Problem 4:

Identify the type of figure shown below. If a small ant is placed inside the shape, can it crawl out without crossing any lines?

A triangle representing a closed figure with a point marked inside.

Solution:

This is a Closed Figure. No, the ant cannot crawl out.

Explanation:

Because the figure starts and ends at the same point, it creates a boundary. In a closed figure, the 'inside' is completely trapped by the lines. Therefore, to move from the inside to the outside, one must cross a line segment.