Area Calculations
RECTANGLES
The area of a rectangle is the product of the base length and its height.
Example: A rectangle whose length is 8 metres has an area of 32 square metres. Calculate the width.
Solution:
Let the width of the rectangle be x
Formula: A = l × w
32 = 8 × x (dividing both sides by 8)
x = 4 metres
SQUARES
A square is a special rectangle where all sides are equal.
Formula: A = side × side
Example: Calculate the area of the square drawn below.
Solution:
Formula: A = s × s
A = 6 cm × 6 cm
Answer: A = 36 cm²
TRIANGLES
The area of a triangle is A = ½(base × height). The base and height always make an angle of 90°.
Example: Calculate the area of the triangle shown below.
Solution:
Formula: A = ½(base × height)
A = ½ (10 × 7)
Answer: A = 35 cm²
TRAPEZIUM
A trapezium is a four-sided figure in which two sides are parallel. The area of a trapezium is given by the formula:
Example:
Find the area of the trapezium shown below.
Solution:
The formula for the area of a trapezium is:
\[ A = \frac{(a + b)}{2} \times h \]
Where:
- a = 10 cm
- b = 8 cm
- h = 5 cm
Now, substituting the values:
\[ A = \frac{(10 + 8)}{2} \times 5 \]
\[ A = \frac{18}{2} \times 5 \]
\[ A = 9 \times 5 = 45 \, \text{cm}^2 \]
CIRCLES
The area of a circle is given by the formula:
Formula: A = π × r² where r is the radius of the circle.
Example: A circle has a diameter of 70 m, calculate the area of the circle.
Solution: The diameter = 70 m, so the radius = 35 m.
Formula: A = π × r²
A = π × (35)²
Answer: A = 3,850 m²
AREA OF IRREGULAR OBJECTS
The steps are:
- Count Full Squares
- Count Partial Squares
- Find the area using the formula:
Formula: Area = ((Number of Full Squares) + (Partial Squares))/2
Example: Find the area of the figure below.
Solution:
The formula for calculating the area is:
\[ A = \text{no. of complete squares} + \frac{\text{no. of incomplete squares}}{2} \]
Where:
- Complete squares = 48
- Incomplete squares = 29
Now, substituting the values:
\[ A = 48 + \frac{29}{2} \]
\[ A = 48 + 14.5 \]
\[ A = 62.5 \, \text{sq units} \]
VOLUME OF REGULAR OBJECTS
The table below shows the formulae used in calculating volumes of cubes, cuboids, and cylinders.
Example: Calculate the volume of a cube of side 5 cm.
Solution:
Volume of cube = s3
Volume = 5 cm × 5 cm × 5 cm = 125 cm3
Example: Calculate the volume of the cuboid measuring 140 cm by 30 cm by 70 cm in cubic meters.
Solution:
To get the answer in cubic meters, we convert all measurements to meters by dividing by 100:
Volume = (140/100) × (30/100) × (70/100) = 0.294 m3
Example: A cylindrical tank has a diameter of 140 cm and a height of 450 cm. Calculate the volume of the tank in cubic centimeters.
Solution:
The diameter is 140 cm, therefore the radius is 70 cm.
Volume of cylinder = π × 702 × 450 = 6 930 000 cm3
Density
Definition
- Density is the mass of a substance per unit volume.
- It tells us how tightly matter is packed in a substance.
- It is a physical property of matter.
Formula:
The formula for density is:
Density (ρ) = Mass (m) / Volume (V)
SI Unit
The SI unit of density is kilogram per cubic meter (kg/m3).
Other common units include:
- grams per cubic centimeter (g/cm3)
- 1 g/cm3 = 1000 kg/m3
Rearranged Formulas
Sometimes, you can calculate mass or volume if density is known:
To calculate Mass, use:
Mass (m) = Density (ρ) × Volume (V)
To calculate Volume, use:
Volume (V) = Mass (m) / Density (ρ)
Example Calculations
Example 1: A substance has a mass of 200 g and a volume of 50 cm3. Find its density.
Solution:
Density = Mass / Volume = 200 g / 50 cm3 = 4.00 g/cm3
Example 2: A block of metal has a volume of 0.02 m3 and a density of 8000 kg/m3. Find its mass.
Solution:
Mass = Density × Volume = 8000 kg/m3 × 0.02 m3 = 160 kg
Determining Density of Irregular Objects Using Eureka Method
1. Principle
The Eureka method (also called the water displacement method) is based on Archimedes' principle: An object submerged in a fluid displaces a volume of fluid equal to its own volume. This method is used when the object does not have a regular shape, so its volume cannot be measured using a formula.
2. Apparatus
- Eureka can (overflow can)
- Measuring cylinder
- Thread or string
- Balance (to measure mass)
- Irregular object
3. Procedure
- Measure the mass of the object using a balance.
- Fill the Eureka can with water until it reaches the spout.
- Suspend the object using a thread and gently lower it into the Eureka can without touching the sides.
- Collect the displaced water in a measuring cylinder.
- Measure the volume of the displaced water. Volume of water displaced = volume of the object.
- Calculate the density using the formula: Density = Mass / Volume.
4. Example: An irregular rock has a mass of 300 g. When submerged in a Eureka can, it displaces 100 cm3 of water.
Solution:
Density = Mass / Volume = 300 g / 100 cm3 = 3.00 g/cm3
5. Key Points / Tips
- The object must be fully submerged.
- Avoid splashing to get an accurate volume measurement.
- Ensure the measuring cylinder measures displaced water accurately.
- This method works for solids that do not dissolve in water.
Speed
1. Definition
- Speed is the distance traveled by an object per unit time.
- It tells us how fast an object is moving, without considering direction.
- It is a scalar quantity (has magnitude only).
2. Formula:
The formula for speed is:
Speed (v) = Distance (d) / Time (t)
3. SI Unit
The SI unit of speed is meter per second (m/s).
Other common units include:
- Kilometers per hour (km/h) → 1 m/s = 3.6 km/h
- Centimeters per second (cm/s)
4. Rearranged Formulas
If speed, distance, or time is unknown, you can calculate it as:
To calculate Speed (v):
v = d / t
To calculate Distance (d):
d = v × t
To calculate Time (t):
t = d / v
5. Example Calculations
Example 1: A car travels 150 km in 3 hours. Find its speed.
Solution:
Speed = Distance / Time = 150 km / 3 hours = 50 km/h
Example 2: A cyclist travels at 5 m/s for 20 seconds. Find the distance covered.
Solution:
Distance = Speed × Time = 5 m/s × 20 s = 100 meters
6. Key Points
- Speed is a scalar (only magnitude, no direction).
- Make sure distance and time units are compatible before calculating.
Summary
Introduction
- Laboratory work involves the use of different apparatus and scientific skills.
- Accurate handling and observation improve experiment results.
- Science learning combines theory and practical activities.
Basic Skills in Science
- Manipulative skills involve handling apparatus correctly.
- Observation skills use senses to gather information.
- Measurement skills require correct units and tools.
- Classification groups items based on similarities.
- Prediction suggests possible outcomes.
- Communication records and shares results.
- Conclusion gives final answers based on findings.
Laboratory Apparatus
- Different apparatus are used for measuring volume, mass, temperature, length, and time.
- Each apparatus has a specific function in experiments.
- Correct use ensures accurate and reliable results.
Care of Apparatus
- Handle apparatus carefully to avoid damage.
- Clean equipment after use.
- Store apparatus properly in safe, dry places.
- Proper care increases durability and accuracy.
Bunsen Burner
- Used for heating substances in the laboratory.
- Has parts such as base, barrel, air hole, collar, and gas inlet.
- Produces yellow (safety) and blue (heating) flames.
Light Microscope
- Used to magnify very small objects.
- Main parts include eyepiece, objective lenses, stage, and adjustment knobs.
- Proper handling and cleaning ensure clear observation.
SI Units
- SI units are standard units used worldwide in science.
- Examples include metre (m), kilogram (kg), and second (s).
- They ensure consistency and accuracy in measurements.
Derived Quantities
- Derived quantities are obtained from basic quantities.
- Examples include area, volume, speed, and density.
- They involve calculations using standard units.
Importance of SI Units
- Ensure uniformity in scientific measurements.
- Help in accurate calculations and comparisons.
- Used globally in science, engineering, and industry.
Practical Activities
- Experiments help apply scientific knowledge.
- Develop skills such as measuring, observing, and recording.
- Improve understanding through hands-on experience.
Assessment
Laboratory Science Assessment
Answer all 20 questions on laboratory apparatus, science skills, SI units, and measurements.
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