Concave Shape in Maths - Definition, Types, Examples
Nikita Aggarwal12 Feb, 2026
What is a Concave Shape?
A concave shape form is a polygon with at least one angle inside it that is more than 180 degrees (a reflex angle). The word "concave" provides you a hint about what it looks like: it looks like one of its sides has "caved in" toward the centre.
When we look at different shapes, we usually tell them apart by their angles and corners. These things must be true for a form to be concave:
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Interior Angles: One of the angles inside the form must be more than 180°.
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Diagonals: At least one diagonal (a line that connects two corners that aren't next to each other) will go outside of the shape.
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Vertices: At least one corner of the polygon points toward the inside.
Definition: The word "concave" means that something curves in, like the inside of a cave. A form is concave if one or more of its sides bend in instead of out.
Different Types of Concave Shapes
Depending on how many sides they have, concave figures look different:
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Different Types of Concave Shapes |
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Intro |
Depending on how many sides they have, concave figures look different: |
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Type 1 |
Darts (quadrilaterals): These are four-sided shapes, one of which points in. It looks like a paper aeroplane that is easier to grasp. |
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Type 2 |
Star polygons: They are a well-known way to make points by "caving in" a lot of vertices. |
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Type 3 |
L-Shapes and T-Shapes: When you write in block letters, the corners of the letters tend to produce hexagons or octagons that are concave. |
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Type 4 |
Cross shapes: A regular plus sign or cross has a concave shape because the arms meet at multiple angles that are reflexive. |
Comparison Between Convex and Concave Shapes
To truly understand a concave shape, it helps to compare it to its opposite: the convex shape. Most geometry shapes taught in early years, such as the rectangle shape, are convex. In a convex polygon, all interior angles are less than 180°, and all vertices point outwards. Here is a quick breakdown of how these kinds of shapes differ:
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Comparison Between Convex and Concave Shapes |
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Intro |
To truly understand a concave shape, it helps to compare it to its opposite: the convex shape. Most geometry shapes taught in early years, such as the rectangle shape, are convex. In a convex polygon, all interior angles are less than 180°, and all vertices point outwards. |
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Transition |
Here is a quick breakdown of how these kinds of shapes differ: |
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Point 1 |
Angle Direction: In a convex shape, all corners point away from the centre. In a concave shape, at least one corner points inward. |
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Point 2 |
Line Test: If you draw a straight line through a convex shape, it will cross the boundary at most twice. If you do the same with a concave shape, the line can cross the boundary four or more times. |
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Point 3 |
The Diagonal Rule: For a rectangle shape or a pentagon, every diagonal stays inside the lines. For concave types of shapes, you can connect two corners with a line that goes "outdoors" before coming back into the shape. |
Read More - Tetrahedron Shape - Meaning, Properties & Formulas
Characteristics of Concave Shape
Not every figure can be a concave shape. Because of the way the angles must work, there are structural rules these geometry shapes must follow:
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Minimum Sides: You cannot have a concave triangle. The simplest concave shape must have at least four sides.
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Complex Boundaries: Because they "cave in," these shapes often look like stars, the letter ‘L’, or arrowheads.
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Irregularity: While regular polygons (like a square) have equal sides and angles, a concave shape is always irregular because its interior angles cannot all be equal if one is reflex.
Examples of Concave Shape with Four Sides
We normally think of a square or a rectangle shape when we talk about shapes with 4 side. These are four-sided shapes. But quadrilaterals can also be concave. People often call a concave quadrilateral a "dart" or a "arrowhead."
In these types of shapes, three of the inside angles are either acute or obtuse, and the fourth one is a reflex angle (larger than 180°). It doesn't appear like a rectangle shape, but the rule still applies: the sum of all the angles inside it is 360 degrees.
Real-World Examples of Concave Shape
We see different shapes in our daily lives that fit this description:
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Star Polygons: A standard five-pointed star is a classic concave shape because the "pits" between the points are reflex angles.
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The Letter 'L': If you trace the outline of a block letter L, you will find one corner that points inward, making it a concave hexagon.
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Arrowheads: Used in signage or tools, the back end of an arrow usually forms a concave angle.
Read More -Octahedron – Definition, Properties & Examples
Solved Examples of Concave Shape for Class 7
Example 1: Figuring Out the Shape
Question: You have a four-sided shape with angles of 30, 40, 50, and 240 degrees inside. Is this concave shape?
Solution:
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Look at the angles; one of them is 240 degrees.
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Since 240 is more than 180, it is a reflex angle.
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In conclusion, this is a concave shape.
Example 2: The Diagonal Test
Question: If you draw a diagonal line between two corners of a shape and that line goes across the space outside the shape, what kind of shape is it?
Solution:
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If a diagonal is outside the boundary, the shape cannot be convex by definition.
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In conclusion, it is a concave shape.
Example 3: Looking at a Rectangle Shape
Question: How many angles that bend back does a normal rectangle have?
Solution:
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A rectangle has four corners, and each one is exactly 90 degrees.
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There are no reflex angles because 90 is less than 180.
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Conclusion: A rectangle is never a shape that curves inward.
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