Area of Parallelograms and Rhombuses

Active tasks make abstract formulas concrete for students. When learners manipulate real objects or visual models, the relationship between base, height, and area becomes visible. For parallelograms and rhombuses, hands-on measuring and cutting reveal why the same area formula applies to both shapes.

• Year 8
• Mathematics
• 3 activities
• 20 min–50 min

Learning Objectives

1. Calculate the area of parallelograms using the formula base times height.
2. Calculate the area of rhombuses using the formula half the product of the diagonals.
3. Explain how a parallelogram can be rearranged into a rectangle of equivalent area.
4. Compare the methods for calculating the area of a parallelogram and a rhombus.
5. Construct a method to find the area of a rhombus given the lengths of its diagonals.

Inquiry Circle

⏱ 40 min·Small Groups

Inquiry Circle: Layering the Prism

Students use MAB blocks to build a prism with a specific base area. They then add layers to see how the volume increases with each 'slice', leading them to the formula V = Base Area x Height.

Prepare & details

Explain how the area formula for a parallelogram relates to the area formula for a rectangle.

Facilitation Tip: During Think-Pair-Share: Prism or Not?, listen for students to describe how equal slices at every level define a prism, correcting any mention of pyramids or cones during the pair discussion.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Simulation Game

⏱ 50 min·Small Groups

Simulation Game: The Packaging Challenge

Students are given a set volume (e.g., 500ml) and must design three different right prisms that could hold that amount. They compare their designs to see which uses the least surface area for the same volume.

Prepare & details

Differentiate between the properties of a parallelogram and a rhombus that affect their area calculations.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Prism or Not?

Students are shown images of various 3D objects (pyramids, spheres, prisms, cones). They discuss in pairs which ones have a 'constant cross-section' and why the volume formula only works for the prisms.

Prepare & details

Construct a method to find the area of a rhombus given its diagonals.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teach this topic by moving from physical slices to abstract formulas. Start with cardstock parallelograms students cut and rearrange into rectangles, then extend the idea to prisms by stacking identical layers. Avoid teaching formulas in isolation; connect each step to the visual transformation students performed. Research shows that students who experience the transformation themselves retain the concept longer than those who only see a demonstration.

What to Expect

By the end of the activities, students will confidently choose the correct formula, justify their choice with clear reasoning, and connect two-dimensional area to three-dimensional volume using cross-sections. They will explain why rearranging a parallelogram into a rectangle does not change the area it covers.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Layering the Prism, watch for students who assume volume requires a rectangular base.

What to Teach Instead

Provide triangular and hexagonal prisms alongside rectangular ones, and ask each group to slice their prism to show identical cross-sections before calculating volume.

Assessment Ideas

Exit Ticket

After Simulation: The Packaging Challenge, give each student a card with a rhombus net and unit cubes. They calculate the area using the net and then the volume of the constructed prism, explaining the connection between the two calculations.

Extensions & Scaffolding

• Challenge students to design a hexagonal prism package with minimal surface area but maximum volume, using grid paper and unit cubes for testing.
• Scaffolding: Provide pre-cut parallelogram shapes with grid lines for students to count squares before applying the formula.
• Deeper exploration: Ask students to derive the rhombus area formula by comparing it to a square with the same diagonals, using a transparency overlay.

Key Vocabulary

Parallelogram	A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal.
Rhombus	A parallelogram with all four sides equal in length. Its diagonals bisect each other at right angles.
Base	For a parallelogram, the base is typically one of its sides. The height is the perpendicular distance from the base to the opposite side.
Height	The perpendicular distance from the base to the opposite side of a parallelogram, or the perpendicular distance between the parallel bases.
Diagonals	Line segments connecting opposite vertices of a polygon. For a rhombus, the diagonals are perpendicular bisectors of each other.