Master

Parallelograms: Area and Perimeter

Master parallelograms: area and perimeter with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Parallelograms: Area and Perimeter

Study Guide

Key Definition

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel and congruent. The area is given by $A = b \, h$, where $b$ is the base and $h$ is the perpendicular distance (height) from that base to the opposite side.

Important Notes

The opposite sides of a parallelogram are congruent: $AB \cong CD$
Opposite angles in a parallelogram are congruent: $\angle A \cong \angle C$
Consecutive angles are supplementary: $\angle A + \angle B = 180^\circ$
The area formula: $A = b \, h$, where $b$ is the length of the base and $h$ is the perpendicular height from the base to the opposite side
The sine–angle area formula: $A = a \, b \, \sin(\theta)$, where $\theta$ is the included angle between sides $a$ and $b$
The perimeter formula: $P = 2(a + b)$, where $a$ and $b$ are the lengths of two adjacent sides

Mathematical Notation

$\angle$ represents an angle

$\cong$ means congruent

$A = b \, h$ is the area formula using base and height

$P = 2(a + b)$ is the perimeter formula

$\sqrt{x}$ represents the square root of $x$

$A = a \, b \, \sin(\theta)$ is the sine–angle area formula

Remember to use proper notation when solving problems

Why It Works

Understanding the properties of parallelograms helps in calculating area and perimeter accurately using the formulas $A = b \, h$, $A = a \, b \, \sin(\theta)$, and $P = 2(a + b)$.

Remember

The area of a parallelogram depends on the base and height: $A = b \, h$. You can also use $A = a \, b \, \sin(\theta)$ when given an included angle.

Quick Reference

Area Formula:$A = b \, h$

Alternate Area Formula:$A = a \, b \, \sin(\theta)$

Perimeter Formula:$P = 2(a + b)$

Understanding Parallelograms: Area and Perimeter

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Given the base $b$ and perpendicular height $h$, the area of a parallelogram is $A = b \, h$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the area of a parallelogram with base $b = 5$ and height $h = 3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A rectangular garden has a length of $10$ meters and a width of $6$ meters. If it is shaped into a parallelogram with the same base and height, what is the area?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a parallelogram has sides $a = 8$ and $b = 6$, and the angle between them is $60^\circ$, calculate the area. (See diagram parallelogram_included_angle.png.)

Challenge Quiz

Single Choice Quiz

Advanced

Calculate the perimeter of a parallelogram with sides $a = 7$ and $b = 9$.

Recap

Watch & Learn

Review key concepts and takeaways