Today we will explore two terms that are often confused but appear in many areas: concave and convex. These words may be new to you, or you may have seen them explained differently.
In this post, we will clarify what makes concave and convex different, explain why they can be confusing, and provide practical examples to help you remember their meanings.
By the end of this post, you should be able to identify concave and convex shapes with confidence. 😉
Index
The Difference Between Concave and Convex
To understand the differences between concave and convex, we must first recognize that these terms are used in different contexts.
- In everyday situations, “concave” describes surfaces or shapes that curve inward or are indented. For example, when you look down into a bowl, the inside surface is concave. “Convex” describes surfaces or shapes that curve outward, like the outside of a ball.
Remember that whether an object is concave or convex can depend on your point of view. For example, if you look at a bowl from above, the inside is concave. If you turn the bowl upside down, the outside becomes convex because it curves outward.
- In mathematics, the terms “concave” and “convex” describe certain properties of shapes, such as polygons, and the curves of functions.
- In mirrors and lenses:
- A concave mirror has a surface that curves inward. These mirrors can focus light and are often used to make images appear larger.
- A convex mirror has a surface that curves outward. This type of mirror causes images to appear smaller and provides a wider field of view.
- Concave lenses are thinner in the center than at the edges. They are often used in eyeglasses to correct nearsightedness, helping people see distant objects more clearly.
- Convex lenses are thicker in the center than at the edges. These lenses correct farsightedness, making nearby objects easier to see.
- In nature, concave shapes include valleys, lakes, and craters on the moon. Convex shapes appear in mountain peaks, hills, and a turtle’s shell.
Concave
Convex
Concavity
Concavity describes a concave shape. This means a surface, shape, or curve is sunken or curves inward, like the inside of a bowl viewed from above.
- Concave surfaces or shapes: A surface is concave when a line drawn between two points on it can extend beyond the shape because the shape is sunken or curved inward.
- Concave curve: For functions, and especially to reach a mathematical consensus on how to name them, we decided to call functions with positive curvature concave curves when viewed from the top of the image.
Convexity
Convexity describes the property of being convex. This means that a surface, curve, or figure bulges or extends outward.
- Convex surfaces or figures: A surface is convex if, when drawing a line between any two points on that surface, the line always lies within the figure. This is clear in circles, but it also holds for regular polygons, which have equal sides and angles.
- Convex curve: As with a concave curve, we agreed to call negative curvature convex when the function forms a convex curve when viewed from the top of the image.
Inflection Point
But what happens when concave and convex sections follow one another? Let’s imagine we’re crossing several hills. If we climb up one side of the first hill, that section is convex; at the top, it remains convex; and as we start descending, it is also convex. But at some point, while we’re going downhill, the shape begins to become concave. That point where it stops being convex and starts to be concave is called the inflection point.
An inflecion point is where a shape or curve changes from concave to convex, or vice versa. In the example below, the curve has two inflecion points.
- At the first point, P₁ (the orange one), the curve changes from convex to concave.
- At the second point, P₂ (the green one), the curve changes from concave to convex.
Practice Activities
Many objects can be classified as concave or convex. Look around your house for some, or draw your own concave and convex objects. But I suggest an experiment. Find a metal spoon and see what happens when you look at your reflection in the concave or convex part. It’s a lot of fun!
After reviewing these examples, you can try related activities from the Smartick sessions to practice identifying concave and convex shapes.
As you’ve probably noticed, we see many concave and convex objects in daily life. If you’d like to learn more about this, you can click on the following link:
If you are interested in practicing and developing your understanding of concave and convex shapes, you can explore additional resources and activities designed to support learning at your own pace.
Sign up for Smartick and take advantage of our personalized online learning method for children ages 4 to 14.
Learn More:
- Learn about Open and Closed Curves
- What Are Solid Geometric Shapes or Bodies?
- Discover Geometric Figures Found in Our Environment
- Trapezium-Trapezoid: Definition, Types and Exercises
- Geometric Shapes with Examples
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A multidisciplinary and multicultural team made up of mathematicians, teachers, professors and other education professionals!
They strive to create the best math content possible.
A multidisciplinary and multicultural team made up of mathematicians, teachers, professors and other education professionals!
They strive to create the best math content possible.
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