Lesson 2: Area

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

For a rectangle with length $L$ and width $W$, the area, $A$, is given by the formula:

Examples

• A rectangular garden has an area of 176 square feet. Its length is 5 feet more than its width. Let width be $w$. The equation is $w(w+5)=176$, or $w^2+5w-176=0$. Solving gives $w=11$. The width is 11 ft and the length is 16 ft.
• The area of a rectangular patio is 250 square meters. The length is twice the width. Let width be $w$. The equation is $w(2w)=250$, or $2w^2=250$. This gives $w^2=125$, so $w = \sqrt{125} \approx 11.2$. The width is about 11.2 m and the length is about 22.4 m.
• A rectangular screen has an area of 90 square inches. Its width is 1 inch less than half its length. Let length be $L$. The equation is $L(0.5L-1)=90$, so $0.5L^2-L-90=0$. This gives $L^2-2L-180=0$. Solving gives $L \approx 14.4$. The length is about 14.4 in and the width is about 6.2 in.

Explanation

This formula helps find unknown dimensions of a rectangle. When length is expressed in terms of width, substituting into the area formula creates a quadratic equation. Solving it reveals the exact measurements for the length and width.