Finding Cone Radius With Given Surface Area And Height Radius Ratio

Jul 12, 2025 by ADMIN 68 views

Unveiling the Radius of a Cone A Mathematical Exploration

In the fascinating world of geometry, cones stand out as elegant three-dimensional shapes with a circular base tapering to a single point called the apex. Understanding the properties and measurements of cones is crucial in various fields, from architecture and engineering to mathematics and design. One essential aspect of cone analysis is determining its surface area, which represents the total area encompassing its curved surface and circular base.

In this article, we embark on a mathematical journey to unravel the relationship between a cone's surface area, height, and radius. We'll delve into a specific problem where the surface area of a cone is given as $216 ext{π}$ square units, and the height is $rac{5}{3}$ times greater than the radius. Our mission is to calculate the length of the radius to the nearest tenth. This exploration will not only reinforce our understanding of cone geometry but also showcase the power of mathematical problem-solving.

Before we dive into the calculations, let's first establish a solid foundation by defining the key concepts involved: surface area, radius, and height of a cone.

Surface Area of a Cone: The surface area of a cone is the sum of the area of its circular base and the area of its curved surface, also known as the lateral surface. The formula for the surface area (SA) of a cone is given by: $SA = πr^2 + πrl$ , where 'r' represents the radius of the base and 'l' denotes the slant height of the cone.
Radius of a Cone: The radius of a cone is the distance from the center of its circular base to any point on the circumference of the base. It is a fundamental parameter that determines the size and shape of the cone.
Height of a Cone: The height of a cone is the perpendicular distance from the apex (the cone's tip) to the center of its circular base. It is another crucial parameter that, along with the radius, defines the cone's dimensions.

With these concepts in mind, we can now proceed to tackle the problem at hand, carefully applying the surface area formula and the given relationship between the height and radius.

Let's begin by clearly stating the problem we aim to solve:

The surface area of a cone is 216π square units. The height of the cone is \frac{5}{3} times greater than the radius. What is the length of the radius of the cone to the nearest tenth?

To solve this problem effectively, we need to extract the crucial information provided. We know two key facts about the cone:

Surface Area: The cone's surface area (SA) is given as $216 ext{π}$ square units. This value will be essential in our calculations as it directly relates to the cone's dimensions.
Height-Radius Relationship: The height (h) of the cone is $\frac{5}{3}$ times greater than its radius (r). This relationship can be expressed mathematically as: $h = \frac{5}{3}r$ . This equation provides a vital link between the cone's height and radius, allowing us to express one in terms of the other.

With this information in hand, we can now formulate a strategy to solve for the radius. The surface area formula involves both the radius and the slant height, but we also have a relationship between the height and radius. This suggests that we can use the Pythagorean theorem to relate the slant height, radius, and height, ultimately allowing us to express the surface area in terms of the radius alone.

Now, let's translate the problem into a mathematical equation and embark on the solution process. Our primary tool will be the surface area formula for a cone:

$SA = πr^2 + πrl$

where:

SA is the surface area
r is the radius
l is the slant height

We are given that $SA = 216π$ . To proceed, we need to express the slant height (l) in terms of the radius (r). This is where the Pythagorean theorem comes into play. In a cone, the slant height, radius, and height form a right triangle, with the slant height as the hypotenuse. Therefore, we have:

$l^2 = r^2 + h^2$

We also know that $h = \frac{5}{3}r$ . Substituting this into the Pythagorean equation, we get:

$l^2 = r^2 + (\frac{5}{3}r)^2$ $l^2 = r^2 + \frac{25}{9}r^2$ $l^2 = \frac{34}{9}r^2$ $l = r\sqrt{\frac{34}{9}}$ $l = \frac{r}{3}\sqrt{34}$

Now we can substitute both the surface area and the expression for 'l' into the surface area formula:

$216π = πr^2 + πr(\frac{r}{3}\sqrt{34})$

We can divide both sides by π to simplify the equation:

$216 = r^2 + \frac{r^2}{3}\sqrt{34}$

Now, let's factor out $r^2$ :

$216 = r^2(1 + \frac{\sqrt{34}}{3})$

To isolate $r^2$ , we divide both sides by the term in the parentheses:

$r^2 = \frac{216}{1 + \frac{\sqrt{34}}{3}}$

Now, we can calculate the value of $r^2$ :

$r^2 ≈ \frac{216}{1 + 1.9437} ≈ \frac{216}{2.9437} ≈ 73.37$

Finally, to find the radius (r), we take the square root of both sides:

$r ≈ \sqrt{73.37} ≈ 8.5656$

Rounding to the nearest tenth, we get:

$r ≈ 8.6$

Therefore, the length of the radius of the cone to the nearest tenth is approximately 8.6 units.

To ensure the accuracy of our solution, let's verify our result by plugging the calculated radius value back into the original equations and checking if the given conditions are satisfied.

We found that the radius (r) is approximately 8.6 units. The height (h) is $\frac{5}{3}$ times the radius, so:

$h = \frac{5}{3} * 8.6 ≈ 14.33$ units

Now, let's calculate the slant height (l) using the Pythagorean theorem:

$l = \sqrt{r^2 + h^2} ≈ \sqrt{8.6^2 + 14.33^2} ≈ \sqrt{73.96 + 205.35} ≈ \sqrt{279.31} ≈ 16.71$ units

Finally, let's calculate the surface area (SA) using the surface area formula:

$SA = πr^2 + πrl ≈ π(8.6)^2 + π(8.6)(16.71) ≈ π(73.96) + π(143.65) ≈ 216π$ square units

The calculated surface area closely matches the given surface area of $216π$ square units, which validates our solution. The small difference can be attributed to rounding errors during the calculations.

In this article, we embarked on a mathematical exploration to determine the radius of a cone given its surface area and the relationship between its height and radius. We successfully navigated through the problem by applying the surface area formula, the Pythagorean theorem, and algebraic manipulation. Our journey culminated in finding the radius of the cone to be approximately 8.6 units.

This problem-solving exercise not only reinforced our understanding of cone geometry but also highlighted the importance of connecting different mathematical concepts to solve complex problems. By combining the surface area formula with the Pythagorean theorem and the given height-radius relationship, we were able to express the surface area in terms of a single variable, the radius, and then solve for it.

The process of verification further solidified our confidence in the solution, demonstrating the power of mathematical reasoning and the interconnectedness of geometric properties. This exploration serves as a testament to the beauty and elegance of mathematics in unraveling the mysteries of the world around us.