Lesson 1.5: Perimeter and Area

New Concept

This lesson explores two fundamental ways to measure geometric figures. We'll define perimeter as the distance around a shape and area as the space it encloses, learning how to calculate each and understanding their distinct applications and units.

What’s next

Next, you'll work through interactive examples to calculate perimeter. Then, you'll tackle practice cards to find the area of various figures.

Property

In mathematics, the perimeter means the length of the boundary of a region. To find the perimeter of a straight-sided figure, measure the length of each side and add up the lengths of the sides.

Examples

• A rectangular garden is 10 meters long and 7 meters wide. Its perimeter is the sum of all its sides: $10 + 7 + 10 + 7 = 34$ meters.
• A triangular park has sides measuring 50 feet, 60 feet, and 75 feet. The perimeter is found by adding these lengths: $50 + 60 + 75 = 185$ feet.
• For a square window with one side measuring 2 feet, the perimeter is $4 \times 2 = 8$ feet, since all four sides are equal.

Explanation

Imagine you are an ant walking around the edge of a shape. The total distance you walk in one full circuit is its perimeter. It measures the length of the boundary, not the space inside.

Property

The number of squares that fit inside a perimeter is called the area of the region enclosed. A square unit is a square that measures 1 unit on each side. Perimeter measures the distance around the outside of a region, while Area measures the amount of space enclosed inside the region.

Examples

• A rectangular piece of paper is 8 inches wide and 11 inches long. Its area is calculated by multiplying the dimensions: $8 \times 11 = 88$ square inches.
• An L-shaped patio is made of two rectangles. One is $4 \text{ m} \times 3 \text{ m}$ and the other is $5 \text{ m} \times 2 \text{ m}$. The total area is $(4 \times 3) + (5 \times 2) = 12 + 10 = 22$ square meters.
• A kitchen floor is 12 feet long and 10 feet wide. The area is $12 \times 10 = 120$ square feet.

Explanation

Area tells you how much surface a shape covers. Think of it as the amount of carpet needed for a floor or paint for a wall. The more square units that fit inside a shape, the larger its area.

Property

There are several methods for calculating the area of a region:

1. Count the unit squares enclosed, including estimates from partial squares.
2. Use multiplication for rectangles ($Area = length \times width$).
3. Enclose the region in a larger rectangle, calculate its area, and subtract the areas of the parts outside the region.

Examples

• A right triangle on a grid with a base of 8 units and a height of 5 units is half of an $8 \times 5$ rectangle. Its area is $\frac{1}{2} \times (8 \times 5) = 20$ square units.
• An octagon is placed inside a $9 \times 9$ square (Area = 81). If the four corner triangles cut off each have an area of 4.5 square units, the octagon's area is $81 - (4 \times 4.5) = 81 - 18 = 63$ square units.
• To find the area of an irregular shape on a grid, you can break it into a $3 \times 4$ rectangle and a $2 \times 5$ rectangle. The total area is $(3 \times 4) + (2 \times 5) = 12 + 10 = 22$ square units.

Explanation

These methods help you find the area of complex or irregular shapes. You can break them into simpler parts or use subtraction to find the space inside, even if the sides aren't all straight or perpendicular.