Finding the Perimeter of Composite Figures
Finding the perimeter of composite figures is a Grade 7 geometry skill in Big Ideas Math, Course 2. A composite figure is formed by combining basic shapes such as squares, rectangles, triangles, and semicircles. The perimeter equals the sum of all outer boundary segments—internal joining edges are excluded. For a square (side 5 cm) with an equilateral triangle attached to one side, the perimeter is 5 + 5 + 5 + 5 + 5 = 25 cm. For a triangle-plus-semicircle, the straight sides are added with the arc length (pi × r). The key strategy: decompose the figure, identify only exterior edges, then sum them using appropriate formulas.
Key Concepts
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 $$P = \text{sum of the lengths of all outer sides}$$.
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.
Common Questions
What is a composite figure in geometry?
A composite figure is a shape made by combining two or more basic geometric shapes such as rectangles, triangles, circles, or semicircles.
Which edges are included when finding the perimeter of a composite figure?
Only the outer boundary segments are included. Internal edges where two shapes are joined together are not part of the perimeter.
How do you find the perimeter of a square with a triangle attached to one side (both with side length 5 cm)?
The perimeter includes three sides of the square (the fourth is interior) plus the two exposed sides of the triangle: 5+5+5+5+5 = 25 cm.
How do you include a semicircle's curved edge in the perimeter?
The arc length of a semicircle is pi × r, where r is the radius. Add this to the straight boundary segments of the composite figure.
What is the general formula for the perimeter of a composite figure?
P = sum of the lengths of all outer sides. Decompose the figure into basic shapes, calculate each exterior segment, and add them all together.
What is the most common mistake when finding perimeter of composite figures?
Including internal edges where shapes connect. Only exterior boundary segments contribute to the perimeter.