CBSE Class 9 Maths Notes: Surface Areas and Volumes

🌐 Surface Area of Sphere & Hemisphere

Let’s dive into the fascinating world of spheres and hemispheres! This section focuses on calculating their surface areas.

📌 Definitions

• Sphere: A three-dimensional object perfectly round, where every point on its surface is equidistant from its center. Think of a ball!
• Hemisphere: Exactly half of a sphere. Imagine cutting a sphere in half.

🔨 Formulaes

• Surface Area of a Sphere: $4\pi r^2$, where ‘r’ is the radius of the sphere.
• Surface Area of a Hemisphere (Curved Surface Area): $2\pi r^2$
• Total Surface Area of a Hemisphere (including the circular base): $3\pi r^2$

Note: Remember that $\pi$ (pi) is approximately 3.14159 or $\frac{22}{7}$.

📚 Volume of Sphere & Hemisphere

Now, let’s explore how to calculate the space occupied by spheres and hemispheres—their volume!

💡 Formulaes

• Volume of a Sphere: $\frac{4}{3}\pi r^3$
• Volume of a Hemisphere: $\frac{2}{3}\pi r^3$

The radius ‘r’ is key for calculating volume, too!

🍆 Surface Area of a Cone

Let’s turn our attention to cones. Here’s how to calculate their surface area.

📌 Definitions

• Cone: A three-dimensional shape that tapers smoothly from a base (usually circular) to a point called the apex or vertex.
• Slant Height (l): The distance from the apex to a point on the circumference of the base.
• Radius (r): The radius of the circular base.

🔨 Formulaes

• Curved Surface Area of a Cone: $\pi r l$, where ‘r’ is the radius and ‘l’ is the slant height.
• Total Surface Area of a Cone: $\pi r l + \pi r^2 = \pi r(l + r)$

Slant height is related to the height (h) and radius (r) by the Pythagorean theorem: $l = \sqrt{r^2 + h^2}$.

🍥 Volume of a Cone

Let’s calculate the space within a cone—its volume.

💡 Formulaes

• Volume of a Cone: $\frac{1}{3}\pi r^2 h$, where ‘r’ is the radius and ‘h’ is the height of the cone.

The height ‘h’ is the perpendicular distance from the apex to the center of the base.

💬 Word Problems

Let’s apply these formulas to solve real-world problems. Here are some examples to get you started:

💡 Problem Solving Tips

• Read Carefully: Understand what the problem is asking.
• Identify Key Information: Note the given values (radius, height, etc.).
• Choose the Right Formula: Select the appropriate formula based on the question.
• Substitute and Calculate: Plug in the values and solve for the unknown.
• Units: Don’t forget to include units (e.g., cm², m³, etc.).

Practice, practice, practice! Working through examples will make you a pro at solving these problems. Good luck!