Compound Shapes: Area and Perimeter

 

What is a Compound Shape?

A compound shape is a shape that is made up of two or more standard 2D shapes (e.g., rectangles, triangles, trapeziums).

To find the area or perimeter, we split the shape into simpler parts.

 

How to Find the Area of a Compound Shape

Steps:

1. Break the shape into standard shapes (e.g., rectangles, triangles, trapeziums).

2. Find the area of each shape using the correct formula.

3. Add the areas together to get the total area.

4. Sometimes subtract areas if necessary (e.g., cutting out sections).

 

Example: Finding the Area of a Complex Compound Shape

 

 

Solution

 

 

• Shape can be split into 3 rectangles and a triangle.

Calculate the area:

$$A_1 = (19 \times 6) = 114$$

$$A_2 = (5 \times 9) = 45$$

$$A_3 = A_2 = 45$$

$$A_4 = \frac{1}{2} \times ( 6 \times 4) = \frac{1}{2} \times 24 = 12$$

$$\text{Total Area} = A_1 + A_2 + A_3 + A_4$$

$$\therefore \quad \text{Total Area} = 114 + 45 + 45 + 12 = 216$$

Final Answer: $216 cm^2$

 

Problem Solving with Area

Many real-world problems involve area calculations, such as:

• Painting a wall

• Laying a carpet

• Designing a sports field

Steps for Problem Solving with Area:

1. Identify the shape in the problem.

2. Look for missing lengths using subtraction or properties of shapes.

3. Find the total area using formulas.

4. Consider cost calculations (e.g., price per square metre).

 

Worked Example: Painting a Playground Wall

Sami is painting a decorative wall at a local playground. The wall is made up of:

• A rectangle that is 4 m wide and 2.5 m tall

• A semicircle sitting on top of the rectangle, with a diameter of 4 m

• Paint A covers $8 m^2$ per tin, costs £19.99 per tin and Paint B Paint costs £2.50 per $m^2$, no partial tins required

Find the total area of the playground and Compare the costs of the paints.

 

 

Find the total area: 

 

 

$$\text{Total Area} = \text{Rectangle Area} + \text{ Sem-circle Area}  = A_1 + A_2$$

$$A_1 = 4 \times 2.5 = 10$$

$$A_2  = \frac{1}{2} \times \pi r^2  \quad \text{Radius} r = \frac{4}{2} = 2$$

$$A_2 = \frac{1}{2} \times \pi(2)^2 = \frac{1}{2}\times \pi \times 4 = 2\pi \approx 6.28$$

Compare Costs Between Two Paints:

• Paint A: $$\frac{16.28}{8} = 2.035 \Rightarrow \text{round up} = 3 \text{ tins }  \\ \text{Total Cost } = 3 \times 19.99 = \pounds 59.97$$

• Paint B: $$\text{Total Cost} = 16.28 \times 2.50 = \pounds 40.70$$

 

Final Answer: Area = $16.28 cm^2$; Cheapest option is Paint B at $\pounds 40.70$