Circumference and Area of Circles and Composite 2D Shapes

These formulas use π (Pi), an irrational constant (approximately $3.14159...$) that represents the ratio of a circle's circumference to its diameter. On calculator papers, always use the $\pi$ button for maximum accuracy. Because circumference is a measure of length, it is always written in linear units (e.g., $cm$, $m$, $mm$).

Worked Example: Calculating Circumference

Calculate the circumference of a circle with a radius of $6\text{ cm}$. Give your answer to 3 significant figures.

Step 1: Select the correct formula based on the given information.

• Since the radius ($r$) is provided, use $C = 2\pi r$.

Step 2: Substitute the value into the formula.

• $C = 2 \times \pi \times 6$
• $C = 12\pi\text{ cm}$

Step 3: Calculate and round to 3 significant figures.

• $C \approx 37.699...\text{ cm}$
• $C = 37.7\text{ cm}$

Area of a Circle

The area is the measure of the total surface inside the boundary of a 2D shape. The formula for the area ($A$) of a circle relies strictly on the radius:

$$A = \pi r^2$$

If a question provides the diameter, you must halve it to find the radius before using the formula. Following the order of operations (BIDMAS), you must square the radius before multiplying by $\pi$. Area is always measured in square units (e.g., $cm^2$, $m^2$).

Worked Example: Calculating Area

Calculate the area of a circle with a diameter of $15\text{ cm}$. Give your answer to 4 significant figures.

Step 1: Find the radius from the diameter.

• $r = 15 \div 2 = 7.5\text{ cm}$

Step 2: Substitute the radius into the area formula, applying BIDMAS.

• $A = \pi \times 7.5^2$
• $A = \pi \times 56.25$

Step 3: Calculate and state the final answer with squared units.

• $A = 56.25\pi\text{ cm}^2$
• $A \approx 176.714...\text{ cm}^2$
• $A=176.7{}^{}$

Perimeters of Semicircles and Sectors

The perimeter of a 2D shape is its total boundary length. For a semicircle or sector, this is not just the curved edge.

A semicircle consists of two parts: the curved arc length (a portion of the circumference) AND the straight edge (the diameter). To find the total perimeter of a semicircle, you must calculate the arc length ($\frac{1}{2}\pi d$) and then add the diameter. Similarly, to find the perimeter of a sector, you calculate the arc length and add the two straight edges (the two radii, $2r$).

Worked Example: Semicircle Perimeter

Calculate the total perimeter of a semicircle with a diameter of $10\text{ cm}$. Give your answer to 2 decimal places.

Step 1: Calculate the curved arc length.

• Arc length $= \frac{1}{2} \times \pi \times d$
• Arc length $= \frac{1}{2} \times \pi \times 10 = 5\pi\text{ cm}$

Step 2: Add the straight edge (diameter) to find the total boundary.

• Total Perimeter $= 5\pi + 10$

Step 3: Calculate the final value.

• Total Perimeter $= 15.707... + 10 = 25.71\text{ cm}$

Areas of Composite Shapes

A composite shape (or compound shape) is made up of two or more standard 2D shapes joined together. To find the area, use the process of decomposition: divide the shape into simpler parts (like rectangles and semicircles), calculate the area of each part individually, and add them together.

If the shape features a "hole" or a shaded region (such as an annulus, which is the ring between two concentric circles), you must use the subtractive method. Calculate the total outer area, calculate the area of the inner "cut out", and subtract the hole from the total.

Worked Example: Area of a Composite Shape

A composite shape is formed by attaching a semicircle to the $6\text{ cm}$ side of a rectangle measuring $8\text{ cm}$ by $6\text{ cm}$. Calculate the total area of the shape to 3 significant figures.

Step 1: Decompose the shape and calculate the area of the rectangle.

• Rectangle area $= 8 \times 6 = 48\text{ cm}^2$

Step 2: Calculate the area of the semicircle.

• The diameter of the semicircle is $6\text{ cm}$, so the radius is $3\text{ cm}$.
• Semicircle area $= \frac{\pi \times 3^2}{2} = 4.5\pi\text{ cm}^2$

Step 3: Sum the areas to find the total.

• Total Area $= 48 + 14.137... = 62.137...\text{ cm}^2$
• Total Area $= 62.1\text{ cm}^2$

Worked Example: Subtractive Method

Calculate the shaded area of a $10\text{ cm}$ by $30\text{ cm}$ rectangle that has a semicircle of diameter $10\text{ cm}$ cut out from it. Give your exact answer in terms of $\pi$.

Step 1: Calculate the total outer area (the rectangle).

• Rectangle Area $= 10 \times 30 = 300\text{ cm}^2$

Step 2: Calculate the area of the hole (the semicircle).

• The radius is $10 \div 2 = 5\text{ cm}$.
• Semicircle Area $= \frac{\pi \times 5^2}{2} = 12.5\pi\text{ cm}^2$

Step 3: Subtract the hole from the total area.

• Shaded Area $= 300 - 12.5\pi\text{ cm}^2$