Let's analyze the image step-by-step to solve the given questions.

Question 8: Write down a formula for the perimeter $P$ mm

The shapes provided consist of a semi-circle attached to a rectangle and a straight edge (triangle-like structure). For each case:

General Formula for Perimeter

The perimeter consists of:

1. The curved part of the semi-circle
   • Length of a semi-circle: $\pi r$, where $r$ is the radius.
2. The two straight edges of the rectangle and triangle-like part.
3. The base of the shape (width of the rectangle).

The total perimeter $P$ is given by: $P = \pi r + \text{straight edges} + \text{base}$

Part (a)

• Given: Radius of semi-circle $r = 20 \, \text{mm}$, and width of the base = $50 \, \text{mm}$.
• Formula: $P = \pi r + 2 \times \text{straight edge} + \text{base}$ Substituting $r = 20$: $P = \pi(20) + 2(40) + 50$

Part (b)

• Given: Radius of semi-circle $r = 40 \, \text{mm}$, base = $a \, \text{mm}$.
• Formula: $P = \pi r + 2 \times \text{straight edge} + a$ Substituting $r = 40$: $P = \pi(40) + 2 \times \text{straight edge} + a$

Part (c) and (d)

• These parts generalize the same pattern using:
  • Radius $r$ = $c \, \text{mm}$ (part d)
  • Base = $a \, \text{mm}$

The general formula: $P = \pi r + 2 \times \text{straight edge} + a$

Question 9: Write down a formula for the area $A \, \text{cm}^2$

The shapes provided are H-shaped, composed of three rectangles. To find the total area $A$, sum the areas of each rectangle.

General Formula

For any H-shaped structure:

1. The total area is given by subtracting the cut-out rectangle areas from the large rectangle: $A = \text{Area of large rectangle} - 2 \times \text{Area of cut-out rectangles}$

Part (a)

• Large rectangle: Width = $3 \, \text{cm}$, height = $8 \, \text{cm}$.
  Area = $3 \times 8 = 24 \, \text{cm}^2$
• Cut-out rectangles: Each cut-out has height $3 \, \text{cm}$, width $2 \, \text{cm}$.
  Area of one cut-out = $2 \times 3 = 6 \, \text{cm}^2$.

$A = 24 - 2(6) = 24 - 12 = 12 \, \text{cm}^2$

Part (b) and (c)

• General Formula:
  Let the large rectangle's dimensions be $m \, \text{cm}$ wide and $8 \, \text{cm}$ high, with cut-out areas.

$A = \text{Area of large rectangle} - 2 \times \text{Area of cut-out rectangles}$

For part b and c, replace $m$ and $n$ into this formula.

Do you want me to explain any part further or provide exact solutions for all steps?

Related Questions

1. How do you calculate the perimeter of a composite shape with circles and straight lines?
2. What is the relationship between area and perimeter for a rectangle?
3. How do you generalize the formula for shapes with multiple components?
4. What is the perimeter of a semi-circular arc combined with a rectangle?
5. How can you simplify formulas for composite shapes?

Tip: Always break down complex shapes into simpler components (e.g., rectangles, triangles, circles) to calculate area or perimeter easily.