Concave Shapes

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Definition: Concave 

The word “concave” means curved inwards or a "cave" inside.

Concave shapes are those shapes in which at least two sides are pushed inwards.

Can you think of any space object that is concave in shape?

A star is a concave shape.

A star is an example of a concave shape.


Different Shapes of Concave

The shapes shown below are concave.

You can observe that the two highlighted sides in each shape are pushed in.

Three different examples of concave shapes with the two sides that are pointing inward highlighted in each shape.

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Concave Examples

When you look at a plane mirror, the image you see is upright and of the same size as yours.

A boy looking at his image in a plane mirror.

You can also see your image on shiny surfaces like on a shiny steel plate or a spoon.

Is the image that you see on a spoon exactly the same as that on a plane mirror? 

Consider the following scenario.

Joe was going to start eating his dinner.

Suddenly, he saw his image on the front side of the spoon.

He observed that the image he saw was different from what he usually sees in a plane mirror.

A man looking at his image on the front side of the spoon observes an upside down image of himself on the spoon. The front side of the spoon is an example of a concave shape.

The reflection on the front side of the spoon was upside down and smaller in size.

Unlike plain mirrors, spoons have curved surfaces.

The front side of a spoon is curved inwards. Such a surface is called concave.

The inside part of a bowl is also an example of a concave surface.

Concave mirrors are used in various medical practices.

For example, a dentist uses a concave mirror to view a relatively larger image of the teeth.

A concave mirror is used to view a larger image of the teeth.

The images on a concave mirror are much larger.

Hence, such mirrors can be used for activities like shaving or applying make-up.

 
Thinking out of the box
Think Tank
  1. Can you think of any concave shape with all its sides equal? If no, what is the reason?
  2. Draw a concave shape and a line segment crossing it.
    Can you tell the minimum number of times this line segment will intersect the boundary of the concave shape?

What is the Difference Between Concave and Convex?

Convex shapes are those shapes in which all the sides are pushed outwards. 

The word “cave” will help you to remember the difference between concave and convex shapes because concave shapes look like they have a cave inside.

Two shapes show the difference between concave and convex shapes. The concave shape has an inward dent or cave.

Concave Shapes Convex Shapes

Concave polygon has atleast one of the interior angles measuring greater than \(180^{\circ}\).

Convex shape has no interior angle that measures more than \(180^{\circ}\).

Concave shape has at least 4 sides.

Convex shape has at least 3 sides.

If a line is drawn inside a concave shape, then it cuts more than two other lines.

If a line is drawn inside a convex shape, it cuts only two other lines.

If you extend the sides of the concave shape, then at least two line segments will intersect each other inside the shape.

If you extend the sides in convex shape, the line segments will not intersect inside the shape.

Time to interact with concavity and convexity!

Drag the vertices of the polygons all around and check if it is concave or convex.

Convex Examples

Let’s get back to the example of Joe and his image on the spoon.

Now, Joe sees his reflection on the back side of the spoon.

His image is not upside down as before but, it appears much bigger than the usual image on a plane mirror.

Such a surface is called convex.

A man sees his image on the back of the spoon. His image is straight as the back side of the spoon is a convex shape.

Stop sign is an example of convex shape and a cross sign is an example of concave shape.

A stop sign is an example of a convex shape and a cross sign is an example of a concave shape.

 
important notes to remember
Important Notes
  1. A polygon is concave if atleast one of the interior angles measures greater than \(180^{\circ}\)

  2. A concave shape has at least 4 sides

  3. If a line is drawn inside a concave shape, then it cuts more than two other lines

  4. If you extend the sides of the concave shape, then at least two line segments will intersect each other inside the shape

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Practice Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

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Frequently Asked Questions (FAQs)

1. What is a concave shape?

Concave shapes are those shapes in which at least two sides are pushed inwards. 

2. What is a concave curve? 

A curve is concave upwards if it turns in the upward direction and concave downwards if it turns in the downward direction.

Different forms of a concave curve that have an upward direction

Different forms of a concave curve that have a downward direction

3. What is a concave function?

If every line segment joining distinct points on the function lies below the graph of the function, that function is said to be a concave function.

Concave and non-concave functions depicted on graph
 

  
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