Area and Volume Problems

1. Essential Information for Solving Problems

When solving problems related to area and volume, following a structured process is key. Here's what you should do:

Understand the Problem: Read the problem carefully to determine what is being asked and which shapes are involved.
Draw a Diagram: If a diagram is not provided, draw one yourself. Clearly label all the given dimensions.
Identify the Shapes: Break down the problem into simpler shapes if it involves a combination of shapes (e.g., a $cylinder$ on top of a $cuboid$ ).
List the Relevant Formulae: Write down the area or volume formulas that apply to the shapes identified.
Check Units: Make sure all measurements are in the same units before performing any calculations.

2. Strategy for Solving Area and Volume Problems

Here's a strategy to follow for systematically solving these problems:

Draw and Label the Diagram:

Start by drawing a diagram that represents the shape or object. Label all known dimensions.

Identify the Shapes Involved:

Recognise the shapes involved (e.g., $cylinder, sphere, cuboid$ ) and note their specific dimensions.

Write Down the Relevant Formula:

Write down the appropriate formula for the area or volume of the shape.

Substitute the Known Values:

Plug the given values into the formula without including units initially.

Perform the Calculation:

Carry out the calculation step by step, ensuring to write out the full expression on each line.

Include Units in the Final Answer:

Finally, include the correct units in your answer (e.g., $cm²$ for area, $cm³$ for volume).

infoNote

Worked Example: Find the Volume of the Cylinder

Problem: Find the volume the cylinder:

Step 1: Identify the Formula

The formula to calculate the volume of a cylinder is: $\text{Volume (V)} = \pi r^2 h$ Where:

$\pi$ is the mathematical constant (use $\pi$ exactly, not as 3.14)
$r$ is the radius of the cylinder's base
$h$ is the height of the cylinder

Step 2: Calculate the Radius

The radius is half of the diameter: $r = \frac{6}{2} = 3$

Step 3: Substitute the Known Values into the Formula

Now, substitute the values for $r$ and $h$ into the volume formula: $\text{V} = \pi \times 3^2 \times 10$

Step 4: Calculate the Square of the Radius

Square the radius: $\text{V} = \pi \times 9 \times 10$

Step 5: Multiply by the Height

Now multiply by the height: $\text{V} = \pi \times 90$

Final Step: Include Units in the Final Answer

Finally, include the units in your answer: $\text{V} = 90\pi \text{ cm}^3$

Final Answer:

The volume of the cylinder is $\mathbf{90\pi \text{ cm}^3}$ .