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# Oblique Cone|Definition & Meaning

## Definition

**Oblique** **cone** is similar to a right **cone** with the exception that the axis of the **cone** does not lie perpendicular to the **base** of the **circular** surface. In other words, the only vertex of the **cone** is not directly centered with the **circular** **base**. A normal right **cone** would look something like a party hat whereas the **oblique** **cone** has a tilted or slanted shape as compared to it predecessor.

A right **cone** would have a **circular** **base** and a closed-shaped axis from its vertex, which would direct towards the center of the **circular** **base,** or the axis would pass through the center of the **base**.

The vertex of the **cone** lies directly above the central point of the **circular** **base**. The reason it is called a right **cone** is because of the angle that the axis makes with the radius of the **circular** **base**. The axis forms a **right angle;** therefore, the term “right” is used.

On the other hand, the **oblique** **cone** also contains a **right angle,** but it is not in line with the central point of the **circular** **base**.

## What Is a Cone?

A **cone** is defined as a **three**–**dimensional** **triangle** having two faces, a **circular** **base**, and a curved face that tightens slickly from a flat **circular** **base** to the vertex (tip of the **cone**). A pyramid is also a kind of **cone** with a triangular cross-section, whereas a **cone** is defined by a **circular** cross-section. Such **cones** having a **circular** cross-section are known as **circular** **cones**.

The unique shape of a **cone** is formed by a set of infinite line segments or the lines that converge at a common point, as we called it **the apex** or **vertex**, and connects this point with all the infinite points on the **circular** **base** circumference.

The perpendicular distance from the vertex of the **cone** to the **circular** **base** is known as the height of the **cone**.

Since **the base** is a circle, the measured values will be of **radius**. In contrast, the tilted distance from the vertex to any point from the infinite points on the **circular** **circumference** is known as the slanted height.

We can measure the area of the **circular** **surface** and the overall volume of the **cone** using these terms. All these terms are shown in figure 2 given above.

We can easily find various **cone**-shaped objects and tools existing in our households, locations we work, workshops, and many more. Some include birthday party **cone**s, waffle ice cream **cone**s, traffic **cone**s, chemical funnels, etc.

## Measurements and Equations of an Oblique Cone

### Slant Height of Cone

As discussed above, the tilted **distance** from the edge of the **cone** to any point on the circumference of the **circular** **base** is known as the **slant** **height**. In the case of an **oblique** **cone**, the vertex does not come perpendicular to the center of the circle, so the height of the **cone** has to be calculated with respect to the same **base**, keeping the angle exactly equal to 90 degrees.

The equation to find the slant height of a regular **cone** or an **oblique** **cone** can be easily derived with the help of the **Pythagoras** **Theorem**.

If we take ‘h’ as the vertical height of the **cone**, L as the slant height, r as the **base** radius, and d as the distance outside the **base**, we get a **right-angled** triangle, thus:

\[ l^2 = (2xr +d)^2 + h^2 \]

Slant Height l = $\sqrt{(2xr +d)^2 +h^2}$

### Cone Volume

A good way to look at **cones** is to visualize a triangle that is being revolved around one of its vertices (preferably the farthest one). The volume of a **cone** can be calculated using a simple example, where we take a **cylindrical** **container** and a **conical** flask having the exact height and the exact **base** radius.

Now start filling up the **conical** **flask** until it is filled to the edge. Upon adding this water to the **container**, you will notice that the container takes up to 3 flasks to fill up. Thus the volume of the container is 3 times that of the volume of the **cone,** or the **cone** volume is one-third of that of the **cylindrical** **container**.

Now it becomes easy to derive the formula, that is, one-third times the area of the **circular** **base** times the height of the **cone:**

V = 1/3 $\times$ **Circular** **base** Area $\times$ **Cone** Height

Since the **area of a circle is equal to $\pi r^2$** and the height of **the cone** is h, so:

\[ V = \dfrac{1}{3} \times \pi r^2 \times h\]

Here **V is the volume** of the **cone**, r is the **base** radius and **h is the vertical height** of the **cone**.

### Surface Area of a Regular Cone

The Total surface area of the **cone** is equal to the **circular** **base** area plus the lateral surface area. The lateral surface area of the **cone** is $\pi rl$ and the surface area of the **base** is $\pi r^2$, thus:

Total Surface Area of the Cone = $\pi rl + \pi r^2$

Area = $\pi rl + \pi (l + r)$

## Elliptic Cone

An elliptic **cone** is a type of **cone** having an elliptical cross-section. The elliptic **cone** is defined by its **directrix** which is in the form of an ellipse. The only difference between this and a regular **circular** **cone** is the **circular** cross-section in its shape. The elliptic **cone **is a quadric surface also known as a degree two **cone **because of its two angles at the vertices.

## Properties of an Oblique Cone

- A
**cone**will only contain one**flat****face**and a round**slanted****face**. - The
**circular****base**narrows down to only one edge called the**vertex**. - A
**cone**can only contain one vertex. - The total volume of a
**cone**is one-third compared to a cylinder of the same**parameters**. - The slant height of a
**cone**is not its actual**height**.

## Frustum of a Cone

When a **cone**, be it a regular **cone** or an **oblique** **cone**, is cut into half by a plane surface, the bottom half of the **cone** is known as the Frustum of **the cone**. The upper half of the sliced **cone** holds the shape of a **cone**, whereas the bottom half gets two **circular** faces with no edges.

This part of the **cone** is obtained when the **cone** is sliced parallel to the **circular** **base**. The volume and the surface area of the frustum of **the cone** cannot be calculated using the formulas of a regular **cone**.

## Solved Example

Find the slant height and volume of an **oblique** **cone** with r = 4cm, h = 7cm, and d = 2cm. Also, find the surface area of the **base **of the** cone**.

### Solution

Using the formula of slant height:

Slant Height, l = $\sqrt{(2xr +d)^2 +h^2}$

Replacing the variables with there values gives us:

\[ l = \sqrt{(2(4) +2)^2 +7^2} \]

\[ l = \sqrt{(10)^2 +49} \]

The slant height of the **oblique** **cone**:

\[ l = \sqrt{149} = 12.2 \text{ cm} \]

Now using the formula for the volume of the **oblique** **cone**,

\[ V = \dfrac{1}{3} \times \pi r^2 \times h\]

Plugging in the values of radius, height, and pi:

\[ V = \dfrac{1}{3} \times 3.142 \times (4)^2 \times 7\]

\[ V = \dfrac{1}{3} \times 3.142 \times 16 \times 7\]

\[ V = 117.2 \text{ cm}^3 \]

Thus the volume of the **cone** comes out to be 117.2 $\text{cm}^3$.

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