Area of Basic Shapes

Area is the amount of space inside a flat (two-dimensional) shape. While perimeter tells you the distance around a shape, area tells you how much surface it covers. Area is measured in square units — square feet ($\text{ft}^2$), square inches ($\text{in}^2$), square meters ($\text{m}^2$), and so on.

Understanding area is essential for estimating materials — flooring, paint, sod, roofing, and fabric are all purchased based on area.

Area of a Rectangle

A rectangle with length $l$ and width $w$ has area:

$A = l \times w$

Example 1: A room is 15 ft long and 12 ft wide. Find the area.

$A = 15 \times 12 = 180 \text{ ft}^2$

Answer: The area is 180 square feet.

Area of a Square

A square is a rectangle where all sides are equal. With side length $s$:

$A = s^2$

Example 2: A square patio tile is 18 inches on each side. Find its area.

$A = 18^2 = 324 \text{ in}^2$

Answer: The area is 324 square inches.

Area of a Triangle

A triangle with base $b$ and height $h$ (the perpendicular distance from the base to the opposite vertex) has area:

$A = \frac{1}{2}bh$

The height must be perpendicular to the base — it forms a right angle with the base. The height is not necessarily a side of the triangle; it may be a line drawn inside or outside the triangle.

Example 3: A triangular piece of land has a base of 40 ft and a height of 25 ft. Find the area.

$A = \frac{1}{2}(40)(25) = \frac{1}{2}(1000) = 500 \text{ ft}^2$

Answer: The area is 500 square feet.

Why does this formula work? A triangle is exactly half of a rectangle with the same base and height. If you duplicate a triangle and flip it, the two pieces form a rectangle with area $b \times h$. Half of that is $\frac{1}{2}bh$.

Area of a Parallelogram

A parallelogram is a four-sided shape where opposite sides are parallel and equal. Its area uses the base $b$ and the perpendicular height $h$ (not the slanted side):

$A = bh$

Example 4: A parallelogram has a base of 10 m and a height of 6 m. Find the area.

$A = 10 \times 6 = 60 \text{ m}^2$

Answer: The area is 60 square meters.

Important: The slanted side of a parallelogram is not the height. The height is always the perpendicular distance between the base and the opposite side.

Area of a Trapezoid

A trapezoid has exactly one pair of parallel sides, called the bases ($b_1$ and $b_2$). The height $h$ is the perpendicular distance between the two bases.

$A = \frac{1}{2}(b_1 + b_2) \times h$

This formula averages the two bases and multiplies by the height.

Example 5: A trapezoid has bases of 8 in and 14 in, with a height of 5 in. Find the area.

$A = \frac{1}{2}(8 + 14)(5) = \frac{1}{2}(22)(5) = \frac{1}{2}(110) = 55 \text{ in}^2$

Answer: The area is 55 square inches.

Area Formula Reference

Real-World Application: Carpentry — Estimating Flooring Materials

A homeowner wants to install hardwood flooring in an L-shaped living room. The room can be divided into two rectangles:

• Rectangle 1: 20 ft by 14 ft
• Rectangle 2: 10 ft by 8 ft

Flooring is sold by the square foot. How many square feet of flooring should be ordered?

Step 1: Find the area of each rectangle.

$A_1 = 20 \times 14 = 280 \text{ ft}^2$

$A_2 = 10 \times 8 = 80 \text{ ft}^2$

Step 2: Add the two areas.

$A_{\text{total}} = 280 + 80 = 360 \text{ ft}^2$

Step 3: Add 10% for waste from cuts, fitting, and defective pieces.

$360 \times 0.10 = 36 \text{ ft}^2$

$360 + 36 = 396 \text{ ft}^2$

Answer: The homeowner should order at least 396 square feet of flooring. Breaking irregular rooms into rectangles (or triangles) is the standard method for estimating area in construction. Always add a waste factor — 10% is typical for flooring, and more complex patterns may require 15%.

Composite Shapes

Real-world shapes are often combinations of basic shapes. To find the area of a composite shape:

1. Break it into basic shapes (rectangles, triangles, etc.)
2. Calculate each area separately
3. Add the areas together (or subtract if a shape is cut out)

Example 6: A wall is 12 ft wide and 9 ft tall, with a triangular gable on top. The gable has a base of 12 ft and a height of 4 ft. Find the total area.

Rectangle (wall):

$A_{\text{rect}} = 12 \times 9 = 108 \text{ ft}^2$

Triangle (gable):

$A_{\text{tri}} = \frac{1}{2}(12)(4) = 24 \text{ ft}^2$

Total:

$A_{\text{total}} = 108 + 24 = 132 \text{ ft}^2$

Answer: The total area is 132 square feet.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Area measures the space inside a flat shape, in square units ($\text{ft}^2$, $\text{m}^2$, etc.)
• Rectangle: $A = lw$. Triangle: $A = \frac{1}{2}bh$. Parallelogram: $A = bh$. Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$
• The height must always be perpendicular to the base — this is a common mistake to watch for
• For composite shapes, break them into basic shapes, find each area, and add (or subtract) as needed
• When ordering materials, always add a waste factor (typically 10%) on top of your calculated area

Return to Geometry for more topics in this section.