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Teacher Guide: Area of Parallelograms

Learn how to calculate the area of a parallelogram using the base and height formula.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Area & Perimeter. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify the base and height of a parallelogram
  • Apply the formula A=b×hA = b \times h to calculate area
  • Distinguish between the height and slant side
  • Solve real-world problems involving parallelogram areas
  • Express area in appropriate square units
Prerequisites
  • • Understanding of multiplication
  • • Knowledge of what perpendicular means
  • • Familiarity with units of measurement
  • • Basic understanding of quadrilaterals
Discussion Starters
  • 1. Why do you think the area formula for a parallelogram is the same as for a rectangle?
  • 2. If you double the height of a parallelogram, what happens to its area?
  • 3. Can two parallelograms with different shapes have the same area? How?
  • 4. Where have you seen parallelogram shapes in buildings or objects around you?
Common Misconceptions

The height is always one of the sides of the parallelogram

Remediation: Draw a parallelogram and show how the height is a separate line segment inside the shape, perpendicular to the base. Use the small square symbol to mark the right angle.

Tilting a rectangle changes its area

Remediation: Demonstrate with paper: cut a rectangle, slice off a triangle from one side, and attach it to the other. The area stays the same even though it's now a parallelogram.

Differentiation Ideas

For Struggling Students:

  • • Start with rectangles and gradually tilt them into parallelograms
  • • Use graph paper to count squares inside the parallelogram
  • • Provide templates with height already drawn and labeled

For On-Level Students:

  • • Find area when height is inside vs. outside the parallelogram
  • • Solve for missing base or height given the area
  • • Compare areas of parallelograms with same perimeter but different heights

For Advanced Students:

  • • Explore how area changes when angles change but sides stay the same
  • • Calculate area using trigonometry (A=absin⁡θA = ab\sin\theta)
  • • Connect to vectors and the cross product
Standards Alignment
  • 6.G.A.1 (CCSS.MATH.CONTENT.6.G.A.1)

    Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes

  • 6.G.A.3 (CCSS.MATH.CONTENT.6.G.A.3)

    Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side

Lesson Resources
  • visualInteractive Parallelogram

    Drag to change base and height, see area update in real-time

  • activityRectangle to Parallelogram

    See how a rectangle transforms into a parallelogram while keeping the same area

  • worksheetFind the Missing Dimension

    Given area and one dimension, find the other

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A parallelogram is a four-sided shape where opposite sides are parallel and equal in length.
To find the area of a parallelogram, we use:
A=b×hA = b \times h
Where:
  • bb is the base (any side of the parallelogram)
  • hh is the height (the perpendicular distance from the base to the opposite side)
Important: The height is NOT the slant side! It must be measured at a right angle (90°) to the base.

Worked Examples

Find the area of a parallelogram with base b=8b = 8 cm and height h=5h = 5 cm.
1

Write the formula

A=b×hA = b \times h → Area formula
2

Substitute the values

A=8×5A = 8 \times 5 → Plug in base and height
3

Calculate

A=40A = 40 → 4040 cm²

Common Mistakes

Using the slant side instead of the perpendicular height

Why it's wrong: The slant side is the length of the tilted edge, not the vertical distance between parallel sides.

Correct: Always look for the height that forms a 90° angle with the base. It's often shown with a small square symbol.

Forgetting to include square units in the answer

Why it's wrong: Area is always measured in square units because we're measuring two-dimensional space.

Correct: If base and height are in cm, the area is in cm². If in meters, the area is in m².

Multiplying all four sides together

Why it's wrong: Unlike perimeter (which uses all sides), area only needs base and height.

Correct: Use only two measurements: A=b×hA = b \times h

Why It Matters

Understanding the area of parallelograms is essential because:
  • Architecture: Parallelogram shapes appear in modern building designs and roof structures
  • Art and Design: Many patterns and tiles use parallelogram shapes
  • Physics: Forces and vectors often form parallelograms (parallelogram law)
  • Foundation for other shapes: The parallelogram formula helps us understand the area of triangles, rhombuses, and other quadrilaterals

Real World Applications

Flooring and Tiles

Some decorative tiles are parallelogram-shaped. Knowing the area helps calculate how many tiles you need.

Example:

A parallelogram tile has a base of 20 cm and height of 15 cm. Each tile covers 20×15=30020 \times 15 = 300 cm² = 0.030.03 m².

1Try It Yourself

You need to cover a floor of 6 m² with parallelogram tiles. Each tile has base 30 cm and height 20 cm.

How many tiles do you need?

Step 1: Write the mathematical expression

First find one tile's area, then divide:

Sail Design

Many sail shapes on boats approximate parallelograms. Calculating sail area is crucial for performance.

Example:

A sail with base 4 m and height 6 m has an area of 4×6=244 \times 6 = 24 m².

2Try It Yourself

A boat needs 30 m² of sail area. The sail base is fixed at 5 m.

What height must the sail be?

Step 1: Write the mathematical expression

Rearrange A=b×hA = b \times h to find hh:

Key Takeaways

  • 1A parallelogram has opposite sides that are parallel and equal
  • 2Area formula: A=b×hA = b \times h (base times height)
  • 3The height must be perpendicular (90°) to the base
  • 4Don't confuse the slant side with the height
  • 5Area is always expressed in square units (cm², m², etc.)

Frequently Asked Questions

Why doesn't the formula use the slant side?

The slant side doesn't tell us the actual vertical space inside the shape. Only the perpendicular height measures the true distance between the parallel sides.

Can I use any side as the base?

Yes! But you must use the height that's perpendicular to that specific base. Different bases have different corresponding heights.

Is the area formula the same as for rectangles?

Yes, it's the same formula (A=b×hA = b \times h). A rectangle is actually a special parallelogram where all angles are 90°.

Glossary

Parallelogram
A four-sided polygon with two pairs of parallel sides
Base
Any side of the parallelogram chosen for calculating area
Height
The perpendicular distance from the base to the opposite side
Perpendicular
At a 90-degree angle
Area
The amount of space inside a two-dimensional shape, measured in square units

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