8.1 Area and Perimeter
Area and Perimeter of Common Shapes
Area is the space inside a 2D shape (measured in square units like cm²).
Perimeter is the distance around the outside of a shape (measured in linear units like cm).
Area is the space inside a 2D shape (measured in square units like cm²).
Perimeter is the distance around the outside of a shape (measured in linear units like cm).
⚡ Essential Formulas:
Triangle
Area = ½ × base × height
Perimeter = sum of all sides
Parallelogram
Area = base × height
Perimeter = 2(a + b)
Trapezium
Area = ½(a + b) × h
Perimeter = sum of all sides
Example 1: Triangle Area
Find the area of a triangle with base 8 cm and height 5 cm.
Formula:
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
Solution:
$\text{Area} = \frac{1}{2} \times 8 \times 5$
$\text{Area} = \frac{1}{2} \times 40$
$\text{Area} = 20$ cm²
Answer: 20 cm²
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
Solution:
$\text{Area} = \frac{1}{2} \times 8 \times 5$
$\text{Area} = \frac{1}{2} \times 40$
$\text{Area} = 20$ cm²
Answer: 20 cm²
Example 2: Parallelogram Area
A parallelogram has a base of 12 cm and perpendicular height of 7 cm. Find its area.
Formula:
$$\text{Area} = \text{base} \times \text{height}$$
Solution:
$\text{Area} = 12 \times 7 = 84$ cm²
Answer: 84 cm²
Note: Don't confuse the slant side with the height! Always use the perpendicular height.
$$\text{Area} = \text{base} \times \text{height}$$
Solution:
$\text{Area} = 12 \times 7 = 84$ cm²
Answer: 84 cm²
Note: Don't confuse the slant side with the height! Always use the perpendicular height.
Example 3: Trapezium Area
A trapezium has parallel sides of 6 cm and 10 cm, and height 4 cm. Find its area.
Formula:
$$\text{Area} = \frac{1}{2}(a + b) \times h$$
where $a$ and $b$ are the parallel sides
Solution:
$\text{Area} = \frac{1}{2}(6 + 10) \times 4$
$\text{Area} = \frac{1}{2} \times 16 \times 4$
$\text{Area} = \frac{1}{2} \times 64$
$\text{Area} = 32$ cm²
Answer: 32 cm²
$$\text{Area} = \frac{1}{2}(a + b) \times h$$
where $a$ and $b$ are the parallel sides
Solution:
$\text{Area} = \frac{1}{2}(6 + 10) \times 4$
$\text{Area} = \frac{1}{2} \times 16 \times 4$
$\text{Area} = \frac{1}{2} \times 64$
$\text{Area} = 32$ cm²
Answer: 32 cm²
💡 Quick Tips:
• Height: Always perpendicular (at 90°) to the base
• Triangle: Half of a rectangle with same base and height
• Parallelogram: Like a rectangle but "pushed over"
• Trapezium: Average the parallel sides, then multiply by height
• Units: Area in cm², perimeter in cm
• Triangle: Half of a rectangle with same base and height
• Parallelogram: Like a rectangle but "pushed over"
• Trapezium: Average the parallel sides, then multiply by height
• Units: Area in cm², perimeter in cm
🎯 Calculate Areas
cm²
cm²
Circle Properties
A circle is a shape where all points are the same distance from the center.
A circle is a shape where all points are the same distance from the center.
⚡ Circle Formulas:
Circumference
$C = 2\pi r$ or $C = \pi d$
Distance around the circle
Area
$A = \pi r^2$
Space inside the circle
Key Terms:
• Radius (r): Distance from center to edge
• Diameter (d): Distance across through center = 2r
• Pi (π): Approximately 3.14159... (use π button on calculator)
• Circumference: Perimeter of a circle
• Radius (r): Distance from center to edge
• Diameter (d): Distance across through center = 2r
• Pi (π): Approximately 3.14159... (use π button on calculator)
• Circumference: Perimeter of a circle
Example 1: Circle Circumference
Find the circumference of a circle with radius 7 cm. Use $\pi = 3.14$
Formula:
$$C = 2\pi r$$
Solution:
$C = 2 \times 3.14 \times 7$
$C = 6.28 \times 7$
$C = 43.96$ cm
Answer: 43.96 cm (or 44 cm rounded)
$$C = 2\pi r$$
Solution:
$C = 2 \times 3.14 \times 7$
$C = 6.28 \times 7$
$C = 43.96$ cm
Answer: 43.96 cm (or 44 cm rounded)
Example 2: Circle Area
A circle has a radius of 5 cm. Calculate its area. Use $\pi = 3.14$
Formula:
$$A = \pi r^2$$
Solution:
$A = 3.14 \times 5^2$
$A = 3.14 \times 25$
$A = 78.5$ cm²
Answer: 78.5 cm²
Formula:
$$A = \pi r^2$$
Solution:
$A = 3.14 \times 5^2$
$A = 3.14 \times 25$
$A = 78.5$ cm²
Answer: 78.5 cm²
Example 3: Finding Radius from Circumference
A circle has a circumference of 31.4 cm. Find its radius. Use $\pi = 3.14$
Formula:
$$C = 2\pi r$$
Solution:
Rearrange: $r = \frac{C}{2\pi}$
$r = \frac{31.4}{2 \times 3.14}$
$r = \frac{31.4}{6.28}$
$r = 5$ cm
Answer: 5 cm
Formula:
$$C = 2\pi r$$
Solution:
Rearrange: $r = \frac{C}{2\pi}$
$r = \frac{31.4}{2 \times 3.14}$
$r = \frac{31.4}{6.28}$
$r = 5$ cm
Answer: 5 cm
💡 Circle Tips:
• Always square the radius for area: $r^2$ means $r \times r$
• If given diameter, divide by 2 to get radius first
• Use the π button on your calculator for accuracy
• If given diameter, divide by 2 to get radius first
• Use the π button on your calculator for accuracy
🎯 Circle Calculations Practice
Parts of a Circle
An arc is part of the circumference. A sector is a slice of the circle.
An arc is part of the circumference. A sector is a slice of the circle.
⚡ Sector and Arc Formulas:
Sector Area
$A = \frac{\theta}{360} \times \pi r^2$
θ is the angle in degrees
Arc Length
$L = \frac{\theta}{360} \times 2\pi r$
Part of the circumference
Think of it as fractions:
If the angle is 90° out of 360°, the sector is $\frac{90}{360} = \frac{1}{4}$ of the whole circle.
If the angle is 90° out of 360°, the sector is $\frac{90}{360} = \frac{1}{4}$ of the whole circle.
Example 1: Sector Area
Find the area of a sector with radius 8 cm and angle 45°. Use $\pi = 3.14$
Formula:
$$A = \frac{\theta}{360} \times \pi r^2$$
Solution:
$A = \frac{45}{360} \times 3.14 \times 8^2$
$A = \frac{45}{360} \times 3.14 \times 64$
$A = 0.125 \times 200.96$
$A = 25.12$ cm²
Answer: 25.12 cm²
$$A = \frac{\theta}{360} \times \pi r^2$$
Solution:
$A = \frac{45}{360} \times 3.14 \times 8^2$
$A = \frac{45}{360} \times 3.14 \times 64$
$A = 0.125 \times 200.96$
$A = 25.12$ cm²
Answer: 25.12 cm²
Example 2: Arc Length
Calculate the arc length for a circle with radius 10 cm and angle 60°. Use $\pi = 3.14$
Formula:
$$L = \frac{\theta}{360} \times 2\pi r$$
Solution:
$L = \frac{60}{360} \times 2 \times 3.14 \times 10$
$L = \frac{60}{360} \times 62.8$
$L = \frac{1}{6} \times 62.8$
$L = 10.47$ cm
Answer: 10.47 cm
Formula:
$$L = \frac{\theta}{360} \times 2\pi r$$
Solution:
$L = \frac{60}{360} \times 2 \times 3.14 \times 10$
$L = \frac{60}{360} \times 62.8$
$L = \frac{1}{6} \times 62.8$
$L = 10.47$ cm
Answer: 10.47 cm
Example 3: Quarter Circle
A quarter circle has radius 6 cm. Find its area and arc length. Use $\pi = 3.14$
Note: Quarter circle means angle = 90°
Sector Area:
$A = \frac{90}{360} \times 3.14 \times 6^2$
$A = \frac{1}{4} \times 3.14 \times 36$
$A = 28.26$ cm²
Arc Length:
$L = \frac{90}{360} \times 2 \times 3.14 \times 6$
$L = \frac{1}{4} \times 37.68$
$L = 9.42$ cm
Answers: Area = 28.26 cm², Arc = 9.42 cm
Note: Quarter circle means angle = 90°
Sector Area:
$A = \frac{90}{360} \times 3.14 \times 6^2$
$A = \frac{1}{4} \times 3.14 \times 36$
$A = 28.26$ cm²
Arc Length:
$L = \frac{90}{360} \times 2 \times 3.14 \times 6$
$L = \frac{1}{4} \times 37.68$
$L = 9.42$ cm
Answers: Area = 28.26 cm², Arc = 9.42 cm
🎯 Sectors and Arcs Practice
Real Life Uses:
• Pizza Slices: Sectors show portion sizes
• Pie Charts: Data visualization using sectors
• Architecture: Curved windows and arches
• Engineering: Gear teeth and circular components
• Sports: Penalty arcs, free throw lines
• Navigation: Compass bearings and coverage areas
• Pizza Slices: Sectors show portion sizes
• Pie Charts: Data visualization using sectors
• Architecture: Curved windows and arches
• Engineering: Gear teeth and circular components
• Sports: Penalty arcs, free throw lines
• Navigation: Compass bearings and coverage areas