What is a Concave Polygon Definition Formula and Key Properties
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 Explained with Properties and Examples
1. What is a concave polygon?
A concave polygon is a polygon with at least one interior angle greater than 180°. This means part of the polygon "caves inward," creating an indentation in its shape.
- It has at least one reflex angle (an angle greater than 180°).
- At least one diagonal lies outside the polygon.
- It differs from a convex polygon, where all interior angles are less than 180°.
2. How do you identify a concave polygon?
A polygon is concave if it has at least one interior angle greater than 180°. You can identify it by checking:
- Measure each interior angle using a protractor.
- If any angle is a reflex angle, the polygon is concave.
- Draw diagonals—if any diagonal lies outside the shape, it is concave.
3. What is the difference between a concave and a convex polygon?
The main difference is that a concave polygon has at least one interior angle greater than 180°, while a convex polygon has all interior angles less than 180°.
- Concave polygon: Has an inward indentation; some diagonals lie outside.
- Convex polygon: No indentations; all diagonals lie inside.
4. What is the formula for the sum of interior angles of a concave polygon?
The sum of interior angles of a concave polygon is given by (n − 2) × 180°, where n is the number of sides. This formula is the same for both concave and convex polygons.
- Example: For a 6-sided concave polygon, sum = (6 − 2) × 180° = 720°.
5. Can a concave polygon have all sides equal?
Yes, a concave polygon can have all sides equal, but it cannot be a regular polygon. A regular polygon must be convex with equal sides and equal angles.
- A concave polygon may have equal side lengths.
- However, at least one angle must be greater than 180°.
6. How do you find the area of a concave polygon?
The area of a concave polygon is found by dividing it into simpler shapes like triangles or rectangles and adding/subtracting their areas.
- Split the polygon into non-overlapping triangles.
- Find each triangle’s area using ½ × base × height.
- Add the areas carefully to get the total.
7. Does a concave polygon have a reflex angle?
Yes, a concave polygon always has at least one reflex angle greater than 180°. This reflex angle creates the inward “dent” in the polygon.
- If no angle exceeds 180°, the polygon is convex.
- The presence of a reflex angle is the defining property of concave polygons.
8. Can a triangle be a concave polygon?
No, a triangle cannot be a concave polygon because all its interior angles are always less than 180°. The sum of angles in a triangle is 180°, so none can be reflex. Therefore, every triangle is always a convex polygon.
9. What are some real-life examples of concave polygons?
Real-life examples of concave polygons include shapes with inward dents or notches.
- An arrowhead shape.
- A star shape (simple star polygon).
- Certain building floor plans with indentations.
10. How many diagonals does a concave polygon have?
The number of diagonals in a concave polygon is given by the formula n(n − 3)/2, where n is the number of sides. This formula is the same for convex polygons.
- Example: For a 5-sided concave polygon, diagonals = 5(5 − 3)/2 = 5.
