The volume of a cone with height equal to radius, and slant height 5 cm is :
Finding the Volume of a Cone with Special Conditions
This question asks us to find the volume of a cone given specific information about its dimensions: the height is equal to the radius, and the slant height is 5 cm.
To solve this, we need to use the formulas related to the dimensions of a cone:
- The relationship between the radius (r), height (h), and slant height (l) of a cone is given by the Pythagorean theorem applied to the right triangle formed by the height, radius, and slant height: \( l^2 = r^2 + h^2 \).
- The formula for the volume (V) of a cone is: \( V = \frac{1}{3} \pi r^2 h \).
Step-by-Step Calculation of Cone Volume
We are given:
- Height \( h \) is equal to the radius \( r \), so \( h = r \).
- Slant height \( l = 5 \) cm.
First, let's use the relationship \( l^2 = r^2 + h^2 \) to find the value of the radius (and height) since \( h = r \).
Substitute the given values into the formula:
\( 5^2 = r^2 + r^2 \)
\( 25 = 2r^2 \)
Now, solve for \( r^2 \):
\( r^2 = \frac{25}{2} \)
We can find \( r \) by taking the square root:
\( r = \sqrt{\frac{25}{2}} = \frac{\sqrt{25}}{\sqrt{2}} = \frac{5}{\sqrt{2}} \)
Since \( h = r \), the height is also \( h = \frac{5}{\sqrt{2}} \) cm.
Next, we need to calculate the volume of the cone using the formula \( V = \frac{1}{3} \pi r^2 h \).
Substitute the values we found for \( r^2 \) and \( h \) into the volume formula:
\( V = \frac{1}{3} \pi \left(\frac{25}{2}\right) \left(\frac{5}{\sqrt{2}}\right) \)
Now, simplify the expression:
\( V = \frac{1}{3} \pi \frac{25 \times 5}{2 \times \sqrt{2}} \)
\( V = \frac{1}{3} \pi \frac{125}{2\sqrt{2}} \)
\( V = \frac{125 \pi}{3 \times 2\sqrt{2}} \)
\( V = \frac{125 \pi}{6\sqrt{2}} \)
The volume of the cone is \( \frac{125 \pi}{6\sqrt{2}} \) cubic centimeters.
Summary of Cone Dimensions and Volume
| Dimension | Value |
|---|---|
| Slant Height (l) | 5 cm |
| Radius (r) | \( \frac{5}{\sqrt{2}} \) cm |
| Height (h) | \( \frac{5}{\sqrt{2}} \) cm (since h=r) |
| Volume (V) | \( \frac{125 \pi}{6\sqrt{2}} \) cm\(^3\) |
Comparing our result with the given options, we find that the calculated volume matches one of the options.
Revision Table: Key Cone Formulas
| Concept | Formula | Description |
|---|---|---|
| Volume of a Cone | \( V = \frac{1}{3} \pi r^2 h \) | Requires radius (r) and height (h). |
| Slant Height Relationship | \( l^2 = r^2 + h^2 \) | Relates slant height (l), radius (r), and height (h). Useful when one dimension is missing or there's a relationship between them. |
| Surface Area (Base) | \( A_{base} = \pi r^2 \) | Area of the circular base. |
| Surface Area (Lateral) | \( A_{lateral} = \pi r l \) | Area of the curved surface. |
| Surface Area (Total) | \( A_{total} = \pi r^2 + \pi r l \) | Sum of base area and lateral area. |
Additional Information on Cones and Geometry
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. The base of a cone is a circle, and the apex is on the line perpendicular to the center of the base.
- Right Circular Cone: The type of cone usually discussed in problems like this one, where the apex is directly above the center of the circular base. The height is the perpendicular distance from the apex to the base.
- Oblique Cone: A cone where the apex is not directly above the center of the base. The formulas for volume and surface area are slightly different or require more complex calculations.
- Key Components: Radius (distance from center of base to edge), Height (perpendicular distance from apex to base), Slant Height (distance from apex to any point on the edge of the base circle). These form a right-angled triangle where the slant height is the hypotenuse.
Understanding these basic concepts and formulas is crucial for solving problems involving the volume and surface area of cones.
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