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A shape can be partitioned or divided into smaller parts in multiple ways. The shares obtained by partitioning a shape have some special properties. Learn about the different types of shares that can be obtained by dividing a shape in multiple ways and the properties associated with them. ...Read MoreRead Less

When a shape is divided into two equal parts, each part is known as half of the original shape. If a shape is divided into three equal parts, each part is known as a third of the original shape. Similarly, if a shape is divided into four equal parts, each part of the shape is known as a fourth, or a quarter, of the shape.

Likewise, most shapes can be divided into halves, thirds, and fourths.

In the case of this triangle, each part is half of the whole, and the whole is made up of two halves.

In the case of this circle, each part is a third of the whole, and the whole is made up of three thirds.

Finally, in the case of this square, each part is a fourth (quarter) of the whole, and the whole is made up of four fourths.

When a shape is divided into equal parts, we get equal shares. In all of the above cases, the shapes have been divided into equal parts or equal shares. That is, each half is equal to each other, each third is equal to the other two thirds, and each fourth is equal to the other three fourths. When these parts are added together, we get the whole shape back.

When a shape is divided into different parts in such a way that the parts are not equal to each other, we get unequal shares.

Unlike equal shares, unequal shares of a shape can be of different shapes and sizes. But unequal shares also add up to the whole.

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**Example 1:** Nicole wants to cut a piece of cloth into two equal parts. The piece of cloth is of the following shape:

In how many ways can Nicole cut this piece of cloth into two equal parts?

**Solution:** Let’s try cutting this shape into two pieces vertically, horizontally, and diagonally.

In the first case, we can see that the line divides the shape into two halves. And both halves have the same shape and size. But in the second and third cases, even though the line splits the shape into two parts, they don’t have the same shape or size. So, Nicole has only one way to divide the piece of cloth into two halves.

**Example 2:** If a circle is cut into two parts, will it always give us two halves?

**Solution:** In general, a circle is divided into two parts by drawing a line through it. Depending on where this line is drawn, a circle can be divided into two parts in multiple ways. Let’s consider some examples.

**Case 1:** The line passes through the center of the circle in all of these cases. It is possible to divide a circle into two halves using this method in as many ways as we want (with slight variations in the line).

**Case 2:** The circles are divided into two parts again in these examples. But here, the line does not pass through the center of the circle. In this case, the two shares of the circle are unequal . That is, they are not of the same shape or size.

Hence, when we divide a circle into two parts, we may not always get two halves.

**Example 3:** Natalie made a caramel pie and a chocolate pie of the same size. She has a fourth of the caramel pie and a third of the chocolate pie. Which of the two slices is a bigger share of the whole pie?

**Solution:**

The caramel pie has to be divided into 3 equal shares or parts such that each slice is a third of the whole pie. The chocolate pie has to be divided into 4 equal shares or parts such that each slice is a fourth or a quarter of the whole pie. Since both pies are the same size, dividing the pies into more slices means each slice would be of a smaller size. This is because the slices have to add up to the whole pie. The chocolate pie has more slices than the caramel pie. Therefore, a third of the caramel pie would be bigger than a fourth of the chocolate pie. It means that the slice of the caramel pie would be bigger than the slice of the chocolate pie.

**Example 4:** Marie painted her wall with four different colors as seen in the figure.

- What share of the wall is painted red?
- What share of the wall is painted purple and green?

**Solution:** The rectangular wall is divided into four equal shares, each with four different colors.

- A fourth or a quarter of the wall is painted red.
- Purple and green together make up half of the entire wall.

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Frequently Asked Questions on Partition of Shapes

We get equal shares when a shape is divided into equal parts, and we get unequal shares when a shape is divided into unequal parts.

For certain shapes, equal shares can be created in multiple ways. Let’s consider a square as an example. The halves of a square can be obtained in the following ways.

In all these cases, the square is divided into equal shares of two halves.

It is not necessary for the equal shares of a figure to have the same shape as the original figure. For example, the equal shares of a circle will not have a circular shape, no matter how we decide to do the partitioning.

Similarly, a square can be divided into equal shares in different ways: along the midpoints and along the diagonals. When a square is divided into four along the midpoints, the shape of each equal share is the same as the original square. But when we divide it along the diagonals, we get equal shares in the shape of triangles.