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# Cross Section|Definition & Meaning

## Definition

The cross-section is defined in geometry as the shape formed by the intersection of a solid and a plane. A three-dimensional cross-section is a two-dimensional geometric shape. In other words, a cross-section is a shape formed by cutting a solid parallel to the base.

## Cross Section

A cross-section represents an object’s **intersection** with a plane along its axis. A cross-section is formed by **cutting a solid** (such as a cone, cylinder, or sphere) with a **plane**.

If a plane parallel cuts a** cylinder-shaped** item to its **base**, the resulting cross-section is a** circle**. As a result, the item has intersected. The object doesn’t need to be three-dimensional; this concept can equally be used for two-dimensional shapes.

You’ll also see some real-life cross-sections, such as a tree after it’s been chopped, which has a ring shape. We get a **square **by cutting a** cubical** **box** with a plane **parallel** to its** base**.

Types of Cross Section

There are two main types of cross-sections

**Cross-section horizontal****Vertical cross-section**

## Cross Section Horizontal

When a plane slices a **solid object** in the** horizontal direction**, it creates a** parallel** cross-section with the **base**, which is known as a** horizontal cross-section**. A cylinder’s horizontal cross-section, for example, is a **circle**.

## Cross Section Vertical

A plane slices the solid shape in the **vertical direction perpendicular** to its base in a **perpendicular cross-section**. A cylinder’s vertical cross-section, for example, is a **rectangle**.

When three-dimensional shapes are cut in various ways, several types of cross-sections result. These can be tough to visualize at times. When a three-dimensional image is chopped **parallel** to its **base**, the shape of the base is obtained. For example, if a cylinder is sliced parallel to its base, the cross-section is a **circle,** much like the cylinder’s base.

Another example would be when cutting a square pyramid parallel to or horizontally to its base. Because the base is also a square, this cross-section would be a square.

## Area of Cross-section

The** **the area of a two-dimensional shape obtained by slicing a t**hree-dimensional object**, such as a cylinder, perpendicular to some defined axis at a point is called **cross-sectional area**.

The cross-section of a cylinder, for example, is a circle. when** sliced parallel** to its base. As a result, the cross-sectional area of this slice is equal to the area of a circle having **radius that is equal to the radius** of the supplied **cylinder**.

Keep in mind how to determine a cylinder’s volume or prism by utilizing its cross-sectional area and length (height), The volume computation is simple if the cross-sectional area is known and **constan**t along the height. But what if the cross-sectional area changes in a predictable way along the height line, as it does for a cone or pyramid, How could a single calculus approach be used to calculate the volume of either of these types of solids?

A circular cylinder can be formed by translating a circular disc along a perpendicular to the disc line. In other words, the cylinder can be formed by moving the cross-sectional area A (the disc) h distance. The resulting volume is known as the volume of the solid and is defined as **V= A x h**.

## Cone Cross Sections

A cutting plane is a plane that forms a cross-section when it intersects a **cone**. A varied shape is generated depending on how the **cutting plane** crosses the cone. Any parallel cross-section of a circular cone generates a** circle** that is comparable (all circles are similar) to the base. This is valid for any parallel cone cross-section.

When a circular cone’s cutting plane is not parallel to its base, the cross-section will be an **ellipse, parabola, or hyperbola**. When the cutting plane to the opposing slant height line is parallel, a **parabola** is created.

## Cylinder Cross Sections

A cylinder’s cross-section can be a** circle**,** rectangle, or oval** depending on how it was cut. The form obtained is a circle if the cylinder has a **horizontal cross-section**. If the plane slices the cylinder **perpendicular** to the base, the resulting shape is a** rectangle**. The oval shape is obtained by cutting the** cylinder parallel** to the base with a minor angle change.

## Sphere Cross Sections

A sphere’s cross-sections are two-dimensional figures created by the** intersection** of a plane with a sphere. Because the sphere has a perfectly round shape with a constant radius, every plane that meets a sphere will form a circular cross-section regardless of the plane’s inclination.

A sphere is a shape that has three dimensions, it is perfectly spherical and has a **constant radius** in all directions. This means that cutting a sphere with a plane will always result in a **circular cross-section**, regardless of the plane’s inclination.

There are three types of cross-sections depending on the direction of the plane that cuts the sphere:

**Cross-section horizontal****Vertical cross-section****Inclined cross-section**

However, as previously stated, all sphere cross-sections will be circular.

## Examples of Cross Section Calculations

### Example 1

what is the cross-sectional area of a plane that is perpendicular to a cube’s base with a volume of 64 cm$^3$.

### Solution

As we already know:

Cube volume = Side$^3$

Therefore:

Side$^3$ = 64 [Given 4 cm on either side]

Because the cube’s cross-section is a square, the side of the square is 4 cm.

As a result, the **cross-sectional area =** **a**$\mathbf{^2}$ **= 4**$\mathbf{^2}$**= 16** **sq.cm**.

### Example 2

Determine the cross-section area of a cylinder with a height of 35cm and a radius of 8 cm?

### Solution

Given:

4 cm radius

25 cm in height

We know that when a plane slices a cylinder parallel to the bottom, the resulting cross-section may be a circle.

As a result, the area of a circle, A, is equal to r$^2$ square units.

Take $\pi$ = 3.14, and change the values:

A = 3.14 (8)$^2$ cm2

A = 3.14 (64) cm$^2$

**A = 200.96 cm**$\mathbf{^2}$

The cylinder’s area is** 200.96 cm**$\mathbf{^2}$.

### Example 3

A solid, metallic right circular cylindrical block with a height of 12 cm and a radius of 9 cm is melted, and little cubes with a 2 cm edge are formed from it. How many cubes may be formed from the block?

### Solution

We have a radius (r) = 9 cm and height (h) = 12 cm for the correct circular cylinder.

As a result:

volume = $\pi$ x r$^2$ x h.

= 3.14 × 9$^2$ × 12 cm$^3$

**= 3052.08 cm$^3$**

*All images/mathematical drawings were created with GeoGebra.*