Question

Geometry Draw a polygon matching each description, if possible. If it is not possible, say so. a concave pentagon

Short answer

Draw a five-sided polygon with one internal angle greater than 180°.

Step-by-step solution

Understand Concave Polygon

A concave polygon is defined as a polygon with at least one internal angle greater than 180° and at least one vertex that points inward.

Understand Pentagon

A pentagon is a five-sided polygon. Therefore, a concave pentagon must have five sides with at least one internal angle greater than 180°.

Draw Five Sides

Start by drawing five connected line segments that form a closed shape. These will be the sides of the pentagon.

Include an Inward-Pointing Vertex

Adjust one of the vertices so that it bends inward, making one of the internal angles greater than 180°.

Check the Shape

Verify that the shape has five sides, one of which is concave, forming an angle greater than 180°.

Key concepts

Concave Polygon

A polygon is a closed figure with straight sides. A concave polygon has a unique feature: at least one of its interior angles is greater than 180°. This causes the polygon to have a 'dent' or an inward-pointing vertex, making it appear as if a part of the shape is 'caved in'.
Understanding this is crucial because it differentiates concave polygons from convex polygons, which have all interior angles less than 180°.
Examples of Concave Polygons:

• A star shape
• A boomerang shape

Concave polygons can be tricky to draw, but remembering the key feature—an angle greater than 180°—helps in identifying and drawing them.

Pentagon Definition

A pentagon is a simple polygon with exactly five sides and five angles. In geometry, naming a polygon by the number of its sides is a common practice; hence, ‘penta’ stands for five. Pentagon shapes can be regular, where all sides and angles are equal, or irregular, where the sides and angles can be of different lengths and degrees.
For the current exercise, you need to focus on an irregular pentagon, specifically one that also meets the concave polygon criteria.
Key Characteristics of a Pentagon:

• Five sides (edges)
• Five vertices (corners)
• Sum of interior angles is 540°

Polygon Angles

The sum of the interior angles of any polygon can be calculated using the formula: \( (n-2) \times 180° \), where \( n \) is the number of sides. For a pentagon, \( n = 5 \), so the sum is \( (5-2) \times 180° = 540° \).
When working with a concave polygon, one of these interior angles will consequently be greater than 180°. For example, in a concave pentagon, if one angle is 210°, the sum of the other four angles must be 330°.
Quick Tips on Angles:

• The sum of exterior angles of any polygon is always 360°.
• Convex polygons have all interior angles less than 180°.
• Concave polygons must have at least one interior angle greater than 180°.

Geometry Drawing

Drawing geometric shapes like a concave pentagon involves step-by-step visualization and adjustment. Here’s how to approach it:
Step-by-Step Guide to Drawing a Concave Pentagon:

• Draw Five Connected Line Segments: Begin by constructing a closed shape with five sides.
• Identify Interior Angles: Ensure one of these angles exceeds 180°. This internal angle will create the 'inward' look.
• Adjust Vertices: Move one vertex inward. Ensure the concavity by checking the angle visually.
• Verify Shape: Reconfirm that you have five connected sides and the required angle greater than 180°.

Remember to use a ruler for straight lines and a protractor to measure angles accurately. Practice makes perfect, so don't hesitate to redraw if necessary.