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# Cone|Definition & Meaning

## Definition

A** cone** is a **three-dimensional geometric** structure that starts from a **circular base** and **converges** as you move up to a point known as the **apex.** It can also be described as a circle that is moving up while continuously **decreasing** in **radius** till it reaches **zero.** The circle part of the cone is known as the **base** and the pointy end of the cone is called the **apex.**

These cones are usually formed by taking a **line segment** from every **point** of the **circumference** and **joining** it at the **apex** point over the circle. This creates a **circular surface area** and shape that is considered a cone.

Geometrically, cones are considered one of the **sub-basic** shapes and are **easy** to **construct.** They bear similarities with **pyramids** with the only difference being that the pyramids have a **quadrilateral base** as compared to a circular one of the cone.

## Properties of Cones

Cones are considered **easy-to-make** shapes that have **properties** that make them **different** from other **geometrical shapes.** A cone will always have **one circular face,** which will be its **base.** It will never be considered a cone shape if the only face it has is not a circle.

Furthermore, having a circular base converging to one point, a cone will **never have an edge,** such as in a pyramid or a cubical shape. The lack of edges in the cone makes it **smooth** and **sleek-looking** with multiple **real-life applications.**

A cone is a **convergence** of **points** of a circle at one point in a different plane, hence it will have a **third property** of having only **one vertex,** that is the **apex.** A pyramid is similar in this regard in that it has one apex point but it also has other vertices on its base due to its **quadrilateral base** structure. Hence, a cone will only have one vertex.

## Elements of a Cone

A cone consists of **three core elements** used to describe the cone’s properties:

- The
**radius**of the**circular base,** **Height**of the**apex**from the base,- The
**slant height**of the**apex**from the**circumference**of the circle.

The **radius** of the **base** decides the **size** of the circle. This is explained to be the **length** of the line from the **circumference** to the **center** of the circle. The greater the radius, the larger will be the base of the cone.

The **height** of the cone is stated as the **perpendicular distance** of the **apex** from the **base** of the cone. This height is not necessarily taken from the center of the circle but it is taken perpendicular to the base straight up to the apex point.

Finally, the **slant height** is the line’s length from the **apex point** to the **outer circumference** of the **circular** **base.** The length of this slanted height can be calculated by using the **Pythagorean theorem:**

l$\mathsf{^2}$ = r$\mathsf{^2}$ + h$\mathsf{^2}$

Here, we have **h** denoting the **height** of the cone, **l** as the **slant height,** and the **radius** of the circular base denoted by **r**.

## Important Formulae Regarding the Cone

A cone has properties that can be calculated using **mathematical formulae: Total volume** and **Surface area.** These two quantities are important to describe the cone’s **implicit values** and **characteristics.**

The **surface area** of the cone is described as the **area** of the **outer shell** of the cone. This includes the **circular base’s** area and the area of the **curved surface** of the cone. The formula for the **curved surface area** is derived as follows:

Curved Surface Area = $\pi$ x l x r

Where **l** is the **slant height, r** is the **radius** and $\pi$ is equal to 3.142. This curved surface area can be added to the area of the circular base $\pi r^2$ to form the following formula of the **total surface area:**

Total Surface Area = $\pi$ x r x l + $\pi$r$\mathsf{^2}$

Hence:

**Total Surface Area =** $\mathbf{\pi}$** x r(r + l)**

The **volume** of the **cone** is the **total amount** of **space** the cone **occupies** or in easier terms, the quantity of liquid it can hold if the cone were to be hollow. The total volume of the cone can be calculated using a formula derived as follows:

Volume = $\dfrac{1}{3} \pi$ x r$\mathsf{^2}$ x h

Here, **r **and **h** respectively represent the **radius** of the **circular base** and the **height** of the cone. This formula is similar to the formula of a **cylinder,** but here we can deduce that the cone is always **one-third of a cylinder** of the **same radius** and **height.**

## An Example of Cones Volume and Surface Area Formulae

A cone is given with a **10 cm height** and a **5 cm radius.** Find the **total surface area** and **volume** of this cone.

### Solution

In this problem, the elements of the cone **h = 10 cm** and** r = 5 cm**. Using this data, we can calculate the **slant height l** of the cone with the help of the **Pythagoras theorem.**

l = $\mathsf{\sqrt{r^2 + h^2}}$

l = $\mathsf{\sqrt{5^2 + 10^2}}$

l = $\mathsf{\sqrt{25 + 100}}$

l = $\mathsf{\sqrt{125}}$

l = 11.18 cm

Thus, after finding the slant height, we can find the **surface area** of the cone as shown below:

Total Surface Area = $\pi$ x 5 (5 + 11.18)

Total Surface Area = $\pi$ x 5(16.18)

Total Surface Area = **254.16** **cm**$\mathsf{^2}$

Hence the **total surface area** is equal to **254.16 square cm.**

Finally, the **volume** of the cone can be calculated as:

Volume = $\dfrac{1}{3} \pi$ x 5$\mathsf{^2}$ x 10

Volume = $\dfrac{1}{3} \pi$ x 25 x 10

Volume = $\dfrac{1}{3} \pi$ x 250

Volume = 261.80 cm$\mathsf{^3}$

Thus, the **volume** of the cone is calculated as **261.8 cubic cm. **

*All drawings and mathematical graphs are made using GeoGebra.*