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# Figures

Have you ever heard someone say they have an hourglass figure? This means they are comparing the shape of their body to that of an hourglass. Some people consider that type of body figure flattering. Well, in geometry, there are different types of figures and shapes and if you continue reading, you will get to know about them.

## Definition of Figures in Geometry

So what is a figure anyway?

A figure is a geometric shape that is a combination of lines, points, or planes that form a closed boundary.

So, when you join two lines or points to create some type of geometric shape, you can call it a figure. Some examples of figures in geometry are shown in the image below.

Examples of geometric figures

If you are new to geometry be sure to check out our article on Fundamentals of Geometry for an introduction.

## Types of Figures in Geometry

Two common types of figures in geometry are:

1. Two-dimensional (2-D) figures; and

2. Three-dimensional (3-D) figures.

### Two-dimensional Figures

A two-dimensional (or 2-D) figure is a flat or plane figure. These kinds of figures are called 2-D because they have just two dimensions, which are length and breadth. Examples of 2-D figures in geometry are squares, rectangles, circles and other polygons. Take a look at the 2-D figures below.

A Triangle; an example of a 2-dimensional figure

A circle; an example of a 2-dimensional figure

You can tell that the figures above are two-dimensional because they are flat shapes on a surface having just length and breadth. You can check out our article on 2-dimensional figures to get more details.

### Three-dimensional Figures

A three-dimensional (or 3-D) figure is a solid and tangible figure. It is tangible because it can be held or picked up. Unlike the 2-D, it's not just a plane figure on a surface. It has depth, volume, and thickness. It is called three-dimensional because it has length, breadth, and depth.

Some examples are cube, cuboid, cylinder, and cone. Some 3-dimensional figures are shown below.

A cube; an example of a 3-dimensional figure

A cylindrical shape; an example of a 3-dimensional figure

The first figure above is a cube figure and the second is a cylindrical figure. You can compare them and the 2-D figures. They do not only have length and breadth, but they also have depth and volume. They are solid figures, and they look like they can be picked up.

## Open and Closed Figures in Geometry

Aside from figures being 2-D or 3-D, there are figures called open and closed in geometry.

An open figure is a figure whose starting point and ending point do not meet.

These kinds of figures are not connected on both ends. It's like having a shape that has a missing part. The figures below are examples of open figures.

An open figure

An open figure

The shapes are either missing a line or a curve, which makes them open figures.

A closed figure is a figure whose start point and end point coincide, that is, they are connected on both ends.

A closed figure is the opposite of an open figure. It's a complete and whole shape. Examples of closed figures in geometry are below.

A closed square figure

A closed triangle figure

Let's look at a quick example.

Identify which of the figures below is an open figure and which is a closed figure.

Solution

Option A is an open figure because its start and end point do not meet. There is a line missing. Option B is a closed figure because its start and endpoint are connected.

## Significant Figures in Geometry

Significant figures are the basic or standard geometric figures. You come across these figures everyday around you. Examples are square, circle, triangle and rectangles.

A square; an example of a significant figure

A Circle; an example of a significant figure

A Rectangle; an example of a significant figure

A Triangle; an example of a significant figure

## Composite Figures in Geometry

Imagine a single figure that can be split or separated into different geometric shapes. This kind of figure is called a composite figure.

A composite figure is a 2-D figure that can be separated into different significant geometric shapes.

You can tell by the name what it means because composite means containing multiple components and that is what a composite figure is.

Take a look at the composite figure below.

A composite figure broken down into a rectangle and triangle

The figure ABCD above is a trapezoid, but it can be separated into two figures. You can see that the trapezoid is made up of a rectangle and a triangle.

In the picture of a kite below, you can see that the figure can be split into two triangles.

A composite figure broken down into two triangles

In the L shape below, you can see that the figure can be split into a square and a rectangle.

A composite figure broken down into a square and a rectangle

All three of the above shapes are composite figures. If you observe closely, you will find out that many polygons in geometry are composite figures.

You may come across a situation where you are asked to find the area or perimeter of a composite figure. To find the area of a composite figure, you will have to find the area of each shape that makes up the composite figure and add them together. This means that you will have to know the formula for finding the area of different shapes.

The same thing applies to finding the perimeter of a composite figure. You find the perimeter by adding the length of each side of the figure.

One of the main uses of composite figures is in determining area and perimeter. When you are asked to find the area or perimeter of a complex figure, you should consider it to be a composite figure. Look at the figure carefully to see the different shapes that it consists of. Find the area of each shape and add them together, or add the sides of every shape present in the case of finding a perimeter. Let's look at a short example.

Find the area and perimeter of the composite figure below.

Solution

To find the perimeter, you have to know the length of all sides of the shape and since it's a composite figure, other sides that may not be visible that you have to consider.

The composite figure above is made up of two rectangles. The broken line in the figure above shows a side that was not visible earlier. Now, you have to find $$x \; cm$$ and $$y\; cm$$ so have the lengths of all the sides to add together.

The long side on the bottom of the figure is $$7 \; cm$$, and the short side on the top of the figure is $$4 \; cm$$. That means $$7 = x + 4$$. Then solving for $$x$$,

\begin{align}x&= 7 - 4 \\ &= 3.\end{align}

Keep in mind you need to include the units, so $$x = 3\; cm$$.

You can see that the long side $$y$$ is actually the sum of the vertical sides opposite it. That means\begin{align}y &= 4+ 2 \\ y &= 6,\end{align}

so $$y = 6\; cm$$.

Therefore, the perimeter will be: $4\; cm + 2\; cm + 4\; cm + 3\; cm + 7\; cm + 6\; cm = 26\; cm.$

To find the area, you will have to find the area of both rectangles present and add them.

The area of A is: \begin{align}\mbox {Area} &= \mbox {length} \cdot \mbox {breadth} \\&= 2 \cdot 4 \\& = 8.\end{align}

Remember that the unit for area is $$cm^2$$ since it is length times width, so the area of A is $$8 \; cm^2$$.

The area of B is:

\begin{align}\mbox {Area} &= \mbox {length} \cdot \mbox {breadth} \\&= 4 \cdot 7 \\& = 28cm^2.\end{align}

Now, you can add the areas of A and B to find the total area:

$\mbox {Total Area} = 8cm^2 + 28cm^2 = 36cm^2.$

## Congruent and Similar Figures in Geometry

If a figure can be flipped or turned to form another figure, then the two figures are congruent.

Congruent figures are figures that have the same size, shape, equal angles and can be positioned differently from each other.

Congruent figures should have corresponding sides and angles. If the figures are not identical in that way, then we can't call them congruent. Examples of congruent figures are below.

A pair of congruent figures

A pair of congruent figures

Take note of the second set of figures above. You might want to say that they are not congruent because they look different. But if you rotate the second one, you will see that it matches the first completely in shape and size.

If you can flip or rotate a figure to match another, then they are congruent figures.

If you want to identify congruent figures, look out for the edges, faces, volume, and even angles. If they are equal or corresponding, then they are congruent. Sometimes, lines are placed on the sides of figures to indicate whether they are congruent or not. See an example below.

A pair of congruent triangles

From the figure above, the sides of the triangles that have just one line are equal in length. The double line and the triple line also mean the same thing.

That means circles are especially easy to check. If two circles have the same radius then they are congruent!

If you are interested in knowing more about congruence, have a look at our article on Congruent Triangles.

Similar figures are figures with proportional corresponding sides, corresponding equal angles, and identical shapes but different sizes.

If the only difference between two figures is size, then we can call them similar. They can have the same shape but are of different sizes.

A pair of similar figures

You can see from the above figures that one is smaller than the other. They look the same and are the same shape, but they are not equal in length, breadth, or volume. So, we call them similar but not congruent.

If a figure can be made to look like another by resizing it, then both figures are similar.

What about circles? Any two circles only really differ by their size, so all circles are similar!

## A cube figure in geometry

A cube is a three-dimensional figure with six equal faces which are all squares.

Sometimes people can mistake a cube for a square, but there is a difference. A cube is a three-dimensional figure, while a square is a two-dimensional figure. In fact, a cube is made up of six squares! The figure below shows a cube and its different parts.

The parts of a cube

Below are some properties of a cube.

1. A cube has six square faces, eight vertices and twelve edges.

2. All the faces of a cube are equal.

3. The opposite faces of a cube are parallel to each other.

4. The opposite edges of a cube are parallel to each other.

## Figures - Key takeaways

• A figure is a geometric shape that is a combination of lines, points, and planes that form a closed boundary.
• An open figure in geometry is a figure whose start and end points are not connected. A closed figure is a figure whose start and end points are connected.
• The difference between congruent and similar figures in geometry is that congruent figures are of the same shape and size, but similar figures are of the same shape but not the same size.
• A composite figure is a 2-dimensional figure that can be separated into two or more figures.

A figure is a geometric shape that is a combination of lines, points and planes that form a closed boundary.

Congruent figures are figures that have the same shape, size and angle.

Types of figures are two dimensional and three dimensional figures.

Examples of figures in geometry are squares, Cubes, circles, triangle, cone, cylinder and many more.

A composite figure is a figure that can be separated into different geometric shapes.

## Final Figures Quiz

Question

What is a figure?

A figure is a geometric shape that is a combination of lines, points and planes that form a closed boundary.

Show question

Question

What are the types of figures in geometry?

1. Two dimensional figures.

2. Three dimensional figures.

Show question

Question

What is a composite figure?

Composite figure is a figure that can be separated into different geometric shapes

Show question

Question

What is the difference between congruent and similar figures?

Congruent figures are of the same shape and size but Similar figures are of the same shape but not the same size.

Show question

Question

Which of these is a composite figure?

A.

B.

C.

A and C

Show question

Question

How many figures are in this composite figure?

There are 3 figures. Two triangles and one rectangle at the middle

Show question

Question

This figure below is a closed figure. True or False.

True

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Question

What is a 2 dimensional figure?

A two dimensional figure is a flat or plane figure that has just two dimensions which are the length and breadth

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Question

What are 3 dimensional figures?

A three dimensional figure is a solid and tangible figure that has its three dimensions as its length, breadth and depth

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Question

The figures below are congruent. True or False.

True

Show question

Question

The figures below are similar. True or False.

True

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Question

Is the figure below a closed figure?

Yes

Show question

Question

What are Similar Figures?

Similar figures are figures that look alike. They have the same shape but not the same size.

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Question

What are Congruent figures?

Congruent figures are figures that are of the same size and shape. They have the same length, width, height and even angles.

Show question

Question

What is the difference between a closed figure and an open figure?

A closed figure is a figure that is connected on both ends, that is, its start and end point meet but an open figure is a figure that is not connected on both ends. The start and the end point of an open figure do not meet.

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