Area of Plane Figures Lesson: Key Concepts & Examples

Lesson Overview

• Understanding Perimeter vs. Area
• What Is Area and How Is It Measured?
• Area of a Square
• Area of a Rectangle
• Labeling Area Correctly
• Measuring Height Correctly
• Understanding the Base of a Figure
• Area of a Parallelogram
• Area of a Triangle
• Area of a Trapezoid
• Solving Area Word Problems

In geometry, we often work with plane figures-these are flat, two-dimensional (2D) shapes like rectangles, squares, triangles, and trapezoids. One of the most important things we can calculate about these figures is their area.

Area is the amount of space inside a shape. It tells us how much surface a shape covers, and it's measured in square units like square centimeters (cm²), square meters (m²), or square inches (in²).

In this lesson, we'll learn:

• What area means and how it's different from perimeter
  
• How to measure area for different plane figures
  
• The correct units and steps for calculating area
  
• Area formulas for squares, rectangles, triangles, parallelograms, and trapezoids
  
• How to apply this knowledge to solve real problems
  

Understanding Perimeter vs. Area

Before diving into area, let's review perimeter so we understand the difference.

• Perimeter is the distance around the outside of a shape. It's like walking along the edge.
  
• Area is the amount of surface inside the shape. It's like painting the inside.
  

Example:

For a rectangle that's 4 cm wide and 6 cm long:

• Perimeter = 4 + 6 + 4 + 6 = 20 cm
  
• Area = 4 × 6 = 24 cm²
  

Key difference: Perimeter is in linear units, while area is in square units​.

What Is Area and How Is It Measured?

Area is measured in square units because it tells us how many unit squares (like 1 cm by 1 cm squares) fit inside a shape.

Always include the correct square unit in your answer!

Take This Quiz:

Area Of Plane Figures Quiz

Area of a Square

A square has four equal sides. The formula to find the area of a square is:

Formula:

Area = side × side
or
Area = side²

Example:

Each side of a square is 3 cm.
Area = 3 × 3 = 9 cm²​

This is one of the simplest area formulas to remember.

Area of a Rectangle

A rectangle has opposite sides that are equal and has four right angles. To find its area, multiply the length and the width.

Formula:

Area = length × width

Example:

Length = 5 in, Width = 4 in
Area = 5 × 4 = 20 in²​

The area tells us how many square units fit inside the rectangle.

Labeling Area Correctly

Always write area answers with the correct square unit.

Example:

If a rectangle is 5 inches by 4 inches:
Area = 20 → write this as 20 square inches or 20 in²​

Writing only "20 inches" is incorrect-it's missing the "square" part.

Measuring Height Correctly

When a formula includes height, like in a triangle or parallelogram, the height must be a straight vertical line.

• It must be perpendicular to the base (forms a 90° angle).
  
• A slanted line does not count as the height​.
  

This rule helps ensure accurate area measurement for angled shapes like triangles and parallelograms.

Take This Quiz:

Area of Plane Figures

Understanding the Base of a Figure

The base is the side of the shape that you measure along the bottom-or sometimes the top.

• In rectangles or squares, it can be any side.
  
• In triangles, it's the side the height touches.
  
• In trapezoids, you use both bases in the formula.
  

So, the base can be the bottom, top, or both-depending on the figure​.

Area of a Parallelogram

A parallelogram has opposite sides that are equal and parallel. Its area is found by multiplying the base by the height.

Formula:

Area = base × height

Example:

Base = 4 m, Height = 7 m
Area = 4 × 7 = 28 m²​

Even though the sides may slant, use only the vertical height in the formula.

Area of a Triangle

A triangle covers half the area of a rectangle or parallelogram with the same base and height.

Formula:

Area = (base × height) ÷ 2

Example 1:

Base = 4 cm, Height = 9 cm
Area = (4 × 9) ÷ 2 = 36 ÷ 2 = 18 cm²​

Example 2:

Base = 20 cm, Height = 25 cm
Area = (20 × 25) ÷ 2 = 500 ÷ 2 = 250 cm²​

Always remember to divide by 2!

Area of a Trapezoid

A trapezoid has one pair of parallel sides. The area formula uses both bases (top and bottom) and the height.

Formula:

Area = (base₁ + base₂) × height ÷ 2

Example 1:

base₁ = 2 in, base₂ = 4 in, height = 3 in
Area = (2 + 4) × 3 ÷ 2 = 6 × 3 ÷ 2 = 18 ÷ 2 = 9 in²​

Example 2:

base₁ = 6 in, base₂ = 10 in, height = 5 in
Area = (6 + 10) × 5 ÷ 2 = 16 × 5 ÷ 2 = 80 ÷ 2 = 40 in²​

It's easy to forget to divide by 2-so be careful!

Solving Area Word Problems

Sometimes you'll use area formulas to solve real-world word problems.

Example:

A triangle has a base of 20 cm and a height of 25 cm. What is its area?

Use the formula: Area = (20 × 25) ÷ 2 = 500 ÷ 2 = 250 cm²​

Always write the unit as square units, based on what measurement was used (cm, m, in...).

Take This Quiz:

PLANE SHAPES Quiz