The total surface area of a cone is the area covered by its surface. This is exactly the part of the cone that is visible when we look at it from the outside. So, how to calculate this area? How to determine how much space is occupied by its surface? And why does it consist of two parts? Let’s figure it out.
Cone Surface Area: Two Important Aspects
A cone is a pyramidal structure. It has a circular base that tapers at the top point known as the apex and is a three-dimensional shape. The area occupied by the perimeter or surface of a cone is called its surface area. The curved surface area and the total surface area of a cone are two different types of surface areas.
Curved surface area of a cone
Curved surface area of a cone is the area that includes only the lateral part of the cone, not including its base. In other words, it is the area of the curved surface of the cone. To calculate this area, we use the following formula:
Total surface area of a cone
Total surface area of a cone is defined as the area occupied by the cone in three-dimensional space, taking into account both the curved surface and the circular base. The formula for calculating this area has the following form:
Note: In these two formulas, R denotes the radius of the base and l is the length of the straight line that connects the center of the base with the top of the cone.
Derivation of the Cone Surface Area Formula: Revealing the Secret
Let’s dive into the derivation of the formula for the area of the complete surface of a cone. Remember how we defined the cone? So, one of the ways of its formation was to cut a sector from the circle and raise it by the radius until the two edges of the sector were connected. Then the radius of the circle became the radius of the cone, and the length of the arc became the length of the circumference of the base.
Now, how do we determine the area of a sector? It’s simple: the length of the arc multiplied by the radius and divided by two. So, if we apply this to our cone, we get:
And the total surface area of the cone is the sum of the area of the base and the сurved surface area. Since the base is a circle with an area of π⋅R2, then for the total surface area we will have:
Thus, we have revealed the secret of this formula!
Total Surface Area of a Cone in Practice: Examples with Answers
The above formulas are used to solve the following examples. Each example has a corresponding solution, but it is recommended that you try to solve them yourself before looking at the answer.
Example 1: What is the curved area of the cone surface with a radius of 5 cm and a slant height of 10 cm?
So, according to the task condition, we have the following values: radius R=5 and slant height l=10. Using the formula for the curved surface area with such values, we get:
Thus, curved surface area of a cone is is157 cm2.
Example 2: What is the total surface area of a cone with a diameter of 12 cm and a slant height of 11 cm?
In this case, we have a diameter instead of a radius. However, we can get a radius by simply dividing the diameter by 2. Therefore, we have: radius R=12/2=6 and the slant height l=11. Substituting these values into our formula, we will have:
Thus, total surface area of a cone is 320.28 cm2.
Example 3: What is the total surface area of a cone with a height of 20 cm and a radius of 8 cm?
So, considering the slant height, the height and the radius of the cone, they form a right triangle, where the slant height is the hypotenuse, the base is the radius of the circular base, and the height is the height of the right triangle.
Using the Pythagorean theorem, we get l2=R2+h2. Thus, the slant height of the cone is equal to the square root of the sum of the squares of the radius and the height of the cone. So, replacing the slant height in the formula for the surface area of the cone with this value, we get:
Further substituting the values h=20 and R=8 in the last expression, we will have:
Thus, total surface area of a cone is 742.061 cm2.
Explore Deeper: Other Aspects of Cone Geometry!
Want to delve even deeper into the world of geometry and math? Here are some fascinating topics related to the study of the cone:
- What is a cone: Definition, parts, examples – Learn more about the structure and main characteristics of a cone, and see practical examples of its application.
- Slant height of a cone: Formula and examples – Consider how to calculate the length of the slant height of a cone using the appropriate formula and applying this concept to examples.
- Volume of a Cone: Formulas and examples – Learn how to calculate the volume of a cone using various formulas and see practical examples for better understanding.
Total Surface Area of a Cone: Flowchart for Quick and Efficient Calculation
