Lesson 6: Perimeter and Area of Composite Figures

Property

To find the perimeter of a composite figure, add the lengths of all the outer boundary segments. Internal lines where shapes are joined are not included in the perimeter. The formula is

Examples

• Square and Triangle: A figure is formed by a square with a side length of 5 cm and an equilateral triangle attached to one side. The perimeter is the sum of three sides of the square and two sides of the triangle: $P = 5 + 5 + 5 + 5 + 5 = 25$ cm.
• Triangle and Semicircle: A right triangle with legs of 6 in and 8 in has a semicircle attached to its hypotenuse (10 in). The perimeter is the sum of the two legs and the arc of the semicircle: $P = 6 + 8 + \frac{1}{2}(\pi \cdot 10) \approx 14 + 15.7 = 29.7$ in.
• Running Track: A running track consists of a rectangle (100 m long, 64 m wide) and two semicircles on each end (diameter 64 m). The perimeter is the length of the two straight sides plus the circumference of one full circle: $P = 100 + 100 + (\pi \cdot 64) \approx 200 + 201 = 401$ m.

Explanation

The strategy for finding the perimeter of a composite figure involves identifying its outer boundary. First, break the figure down into its basic shapes, like rectangles, triangles, and circles. Next, calculate the length of each segment that forms the exterior of the figure, using formulas like circumference for curved parts. Finally, sum these exterior lengths to find the total perimeter, making sure to exclude any lines that are internal to the composite shape.

Property

On grid paper, estimate perimeter by counting unit lengths along grid lines as $1$ unit each and diagonal lengths as $1.5$ units each. Total estimated perimeter = (number of unit lengths × $1$) + (number of diagonal lengths × $1.5$).

Examples

Property

There are several methods for calculating the area of composite figures:

1. Count the unit squares enclosed, including estimates from partial squares.
2. Use multiplication for rectangles ($Area = length \times width$).
3. Break the composite figure into simpler shapes (rectangles, triangles, circles) and add their areas together.

Examples