The radius of a hemisphere is 5 cm. Find the volume.
261.9 cm3
Finding the Volume of a Hemisphere Given Its Radius
This problem asks us to calculate the volume of a hemisphere when its radius is known. A hemisphere is exactly half of a sphere.
The formula for the volume of a sphere is \(\frac{4}{3}\pi r^3\), where \(r\) is the radius. Since a hemisphere is half of a sphere, the formula for the volume of a hemisphere is half of the sphere's volume.
The volume of a hemisphere is given by the formula:
\(V = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3\)
Applying the Hemisphere Volume Formula
We are given that the radius of the hemisphere is 5 cm.
- Radius, \(r = 5\) cm
Now, we can substitute the value of the radius into the hemisphere volume formula:
\(V = \frac{2}{3}\pi (5 \text{ cm})^3\)
\(V = \frac{2}{3}\pi (125 \text{ cm}^3)\)
\(V = \frac{250}{3}\pi \text{ cm}^3\)
To get a numerical value, we use an approximate value for \(\pi\), such as 3.14159.
\(V \approx \frac{250}{3} \times 3.14159 \text{ cm}^3\)
\(V \approx 83.333 \times 3.14159 \text{ cm}^3\)
\(V \approx 261.799 \text{ cm}^3\)
Comparing Calculated Volume with Options
Our calculated volume is approximately 261.799 cm³. Let's look at the given options:
- Option 1: 330.4 cm³
- Option 2: 117.9 cm³
- Option 3: 150.6 cm³
- Option 4: 261.9 cm³
The calculated volume 261.799 cm³ is very close to 261.9 cm³. The difference is likely due to rounding the value of \(\pi\).
Conclusion on Hemisphere Volume
Based on the calculation using the volume of a hemisphere formula and the given radius of 5 cm, the volume is approximately 261.8 cm³, which matches option 4 when rounded to one decimal place.
Revision Table: Solid Geometry Formulas
| Shape | Volume Formula | Surface Area Formula |
|---|---|---|
| Sphere | \(\frac{4}{3}\pi r^3\) | \(4\pi r^2\) |
| Hemisphere (Solid) | \(\frac{2}{3}\pi r^3\) | \(3\pi r^2\) (Curved surface + Base area) |
| Cylinder | \(\pi r^2 h\) | \(2\pi r(r+h)\) |
| Cone | \(\frac{1}{3}\pi r^2 h\) | \(\pi r(r+l)\), where \(l\) is slant height |
Additional Information on Hemispheres and Spheres
A hemisphere is a three-dimensional shape formed by cutting a sphere into two equal halves along its diameter. It has a curved surface and a flat circular base.
- The radius (\(r\)) of the hemisphere is the same as the radius of the original sphere and the radius of the circular base.
- The height of the hemisphere is equal to its radius.
- The curved surface area of a hemisphere is half the surface area of the sphere, which is \(2\pi r^2\).
- The total surface area of a solid hemisphere includes the curved surface area and the area of the flat circular base (\(\pi r^2\)), totaling \(2\pi r^2 + \pi r^2 = 3\pi r^2\).
- Volume is a measure of the space occupied by a three-dimensional object, measured in cubic units like cm³.
Understanding the relationship between spheres and hemispheres is key to calculating their volumes and surface areas correctly.
Important Questions from Solid Figures
- Find the volume of the cylinder whose radius is 38.5 cm and height is 18 cm. (use $\pi = 22/7$ and In $cm^3$)
- Find the surface area of the sphere (in $cm^2$) whose radius is 168 cm and $\pi = 22/7$.
- Find the volume of the cylinder whose radius is 21 cm and height is 13 cm. (In $cm^3$ and use $\pi = 22/7$)
- A solid spherical ball made of iron is melted and two new balls are made whose diameters are in the ratio of 1:2. The ratio of the volume of the smaller new ball to the original ball is
If a cubical container of length, breadth and height each of 10 cm can contain exactly 1 litre of water, then a spherical container of radius 10.5 cm can contain
