What is a Concave Polygon in Maths?
The concept of concave polygon plays a key role in mathematics and geometry. It helps students classify polygons, understand properties about angles and diagonals, and spot differences from convex polygons in visual problems and exams.
What Is a Concave Polygon?
A concave polygon is a closed, flat shape with straight sides in which at least one interior angle is greater than 180°. This causes the polygon to have an “indentation” or a part that “caves in.” Some of the diagonals in a concave polygon may also lie outside the shape. You’ll find this concept applied when classifying types of polygons, identifying shapes with reflex angles, and in geometry-based reasoning questions.
Key Formula for Concave Polygon
Here’s the standard formula for the sum of interior angles of any n-sided polygon, including a concave one: Sum of interior angles = \( (n - 2) \times 180^\circ \) For example, a concave hexagon (6 sides) will have a total interior angle sum of \( (6-2) \times 180^\circ = 720^\circ \).
Properties of Concave Polygon
- At least one interior angle is a reflex angle (greater than 180° and less than 360°).
- May have more than one reflex interior angle.
- At least one diagonal lies outside the polygon.
- Some vertices “point inwards” towards the interior.
- Always an irregular polygon (cannot be regular).
- Can be split into a set of convex polygons for area calculation.
- Sum of exterior angles is always 360°.
- The sum of interior angles depends only on the number of sides and is given by the same formula as convex polygons.
Concave Polygon Examples
- Dart (arrowhead) quadrilateral: A 4-sided polygon with an inward-pointing angle.
- Star (pentagram): A classic example where several angles “cave in.”
- Concave hexagon: A 6-sided figure with at least one interior angle above 180°.
- Polygon with an indented side: Any multi-sided shape where you notice a “dent” or interior-pointing vertex.
Many real-life objects, such as some arrowheads or decorative stars, are concave polygons. For clear visuals, see standard diagrams in Vedantu’s live classes or worksheets for better understanding.
Table: Concave vs Convex Polygon
| Feature | Concave Polygon | Convex Polygon |
|---|---|---|
| Interior Angles | At least one angle > 180° | All angles < 180° |
| Diagonals | Some diagonals lie outside the polygon | All diagonals lie inside |
| Vertices | Some point inwards | All point outwards |
| Regularity | Always irregular | Can be regular or irregular |
| Sides | Minimum 4 sides | Minimum 3 sides |
Step-by-Step Illustration: Area and Perimeter of a Concave Polygon
Suppose a concave polygon is made by joining a rectangle and a square as below.
1. Find the area of the rectangle (length = 24 units, width = 10 units):2. Area of rectangle = 24 × 10 = 240 sq units
3. Find the area of the square (side = 8 units):
4. Area of square = 8 × 8 = 64 sq units
5. Total area of the concave polygon = 240 + 64 = 304 sq units
6. To find the perimeter, add all the outer side lengths.
7. Example side lengths: 24, 18, 8, 8, 16, 10
8. Perimeter = 24 + 18 + 8 + 8 + 16 + 10 = 84 units
Tip: Divide complex concave polygons into basic shapes, find individual areas, then sum them up for the answer.
Relation to Other Concepts
Understanding concave polygons helps with classifying polygons, understanding angle sums, and identifying regular polygons (which can never be concave). It connects directly to the study of types of angles, especially reflex angles, and polygon angle sum properties.
Frequent Errors and Misunderstandings
- Mixing up convex and concave polygons due to visual confusion.
- Thinking that a triangle can be concave—remember, it’s not possible!
- Forgetting that regular polygons are always convex, never concave.
- Assuming that sides must be unequal—only the angle property matters.
Classroom Tip
A quick way to remember a concave polygon: “If you can find an angle that makes the shape look like it’s caving in, or a diagonal that goes outside the polygon, it’s concave!” Vedantu’s teachers use this quick check in live classes and worksheets for instant identification.
Try These Yourself
- Draw a polygon with at least one angle > 180° and check if it’s concave.
- Is a star shape a concave polygon?
- Find the sum of interior angles of a concave hexagon.
- Give two real-life examples of concave polygons.
Wrapping It All Up
We explored concave polygons—from definition, formula, properties, examples, common mistakes, and their connections. Practice spotting and drawing them for exams using Vedantu’s worksheets, and you’ll master this geometry concept in no time.
See also: Convex Polygon, Types of Polygons, Polygon Angle Sum Property
FAQs on Concave Polygon: Definition, Properties, and Examples
1. What is the difference between concave and convex polygons?
A convex polygon has all its interior angles less than 180°. All its diagonals lie entirely within the polygon. In contrast, a concave polygon has at least one interior angle greater than 180°, and at least one diagonal lies outside the polygon. Think of it as having at least one 'dent' or inward-pointing section.
2. How do I identify a concave polygon?
To identify a concave polygon, look for these key features:
• At least one interior angle measuring greater than 180° (a reflex angle).
• A noticeable 'dent' or inward curve in the shape.
• At least one diagonal that lies outside the polygon itself.
3. Can a polygon have more than one concave angle?
Yes, a concave polygon can have multiple interior angles greater than 180°. The defining characteristic is having at least one such angle.
4. What are some examples of concave polygons?
Examples include a concave quadrilateral (like a dart or arrowhead), certain star shapes, and some irregular polygons with more than four sides. Imagine a square with one corner pushed inwards—that's a concave polygon.
5. What is the sum of angles in a concave polygon?
The sum of the interior angles of any polygon (concave or convex) is calculated using the formula: (n - 2) * 180°, where 'n' is the number of sides. This formula holds true regardless of whether the polygon is concave or convex.
6. How to calculate the area of a concave polygon?
There's no single formula for the area of a concave polygon. The approach is to divide the concave polygon into smaller, simpler shapes (triangles, rectangles, etc.) whose areas you can easily calculate. Then, sum the areas of these smaller shapes to find the total area of the concave polygon.
7. How to calculate the perimeter of a concave polygon?
The perimeter of a concave polygon is simply the sum of the lengths of all its sides. Measure each side and add the lengths together to find the total perimeter.
8. What is a regular concave polygon?
A regular polygon has all sides and angles equal. By definition, a concave polygon has at least one angle greater than 180°. Therefore, a regular polygon can never be concave. All regular polygons are convex.
9. Why do some diagonals of a concave polygon lie outside the shape?
Because of the presence of reflex angles (angles > 180°), some diagonals connecting vertices will necessarily extend beyond the polygon's boundary. This is a direct consequence of the inward-curving sides characteristic of concave polygons.
10. Can a triangle be a concave polygon?
No, a triangle cannot be concave. The sum of the interior angles of a triangle is always 180°, meaning no angle can be greater than 180°. Concave polygons require at least one reflex angle.
11. What are some real-life examples of concave polygons?
Concave shapes appear in various places. Think of a boomerang, certain types of star shapes, or even the outline of a weirdly shaped piece of land. Many everyday objects might contain concave polygonal sections within their overall form.
12. How is the formula for the sum of interior angles applied to concave polygons?
The formula (n-2) * 180° applies to concave polygons just as it does to convex polygons. Even though some angles are reflex, the overall sum remains consistent based on the number of sides (n).
