Mathematics · Class 10

Active learning ideas

Areas of Combinations of Plane Figures

Active learning works well for areas of combinations of plane figures because students often struggle to visualise how multiple shapes interact in a single diagram. When they cut, measure, and reassemble paper models, they build a clearer mental map of the components, which is essential for accurate calculations in mensuration problems.

• CBSE Learning Outcomes
• NCERT: Areas Related to Circles - Class 10

• 35–50 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Pairs

Paper Cutouts: Compose and Shade

Provide students with printed shapes like semicircles, rectangles, and triangles on cardstock. Instruct them to cut, arrange into a complex figure, shade a region, and calculate its area by adding or subtracting components. Pairs swap designs to verify each other's work.

Differentiate between adding and subtracting areas when dealing with combined figures.

Facilitation Tip

During Paper Cutouts, ask students to label each piece with its area formula and value before reassembling, so they connect the physical action to the mathematical process.

What to look for

Present students with a diagram of a square with a semicircle attached to one side. Ask them to write down the formulas they would use to find the total area and identify which shapes need to be added.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 40 min · Small Groups

Shaded Relay: Team Breakdown

Display a large diagram of a combined figure on the board. Divide class into teams; each member solves one part (e.g., sector area), passes to next for addition/subtraction. First team with correct total shaded area wins. Debrief strategies used.

Design a strategy to calculate the area of a complex shaded region.

Facilitation Tip

In Shaded Relay, circulate and listen for groups explaining their subtraction steps aloud, as verbalising the logic helps solidify understanding.

What to look for

Provide a complex shaded region (e.g., a circle with a smaller circle removed from its center, or a rectangle with a quarter-circle at one corner). Ask students to calculate the area and briefly explain their strategy for dealing with the shaded portion.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 45 min · Individual

GeoGebra Exploration: Dynamic Figures

Students use free GeoGebra software to draw circles and sectors, overlay polygons, and shade regions. Adjust parameters to see area changes, then compute manually and compare. Share screenshots in a class gallery for peer review.

Evaluate the most efficient method for finding the area of a figure composed of multiple basic shapes.

Facilitation Tip

For GeoGebra Exploration, model how to drag vertices to test edge cases, such as when a semicircle’s diameter aligns with a rectangle’s side.

What to look for

Show two different methods for calculating the area of a figure with overlapping circular regions. Ask students to discuss which method is more efficient and why, focusing on the steps involved in subtraction.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 50 min · Small Groups

Park Design Challenge: Whole Class Project

Groups sketch a park with paths, ponds (sectors), and lawns (combined shapes), calculate total area and shaded pathways. Present to class, justifying methods. Vote on most efficient design.

Differentiate between adding and subtracting areas when dealing with combined figures.

Facilitation Tip

In Park Design Challenge, insist on a shared legend for symbols, so students communicate their design choices clearly to peers.

What to look for

Present students with a diagram of a square with a semicircle attached to one side. Ask them to write down the formulas they would use to find the total area and identify which shapes need to be added.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Teachers should begin with simple combinations, like a rectangle with a semicircle, before moving to overlapping circles or irregular polygons. Avoid rushing to formulas; instead, emphasise estimation and visual decomposition first. Research shows that students who draw and cut their own figures retain procedural steps longer than those who only observe demonstrations. Encourage frequent peer teaching, as explaining to others reveals gaps in reasoning.

Successful learning looks like students confidently dissect complex figures into basic shapes, apply the correct area formulas with precision, and justify their addition or subtraction steps with clear reasoning. By the end of the activities, they should be able to explain why a shaded region requires subtraction, not just addition, even when parts overlap or extend beyond standard shapes.

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Watch Out for These Misconceptions

• During Paper Cutouts, watch for students who add all visible areas without considering overlaps or shared boundaries.

  Ask them to physically remove the overlapping part from one shape and place it aside, then recount the total area. Circulate and guide them to write subtraction steps for the removed area.

• During Shaded Relay, watch for students who divide the circle’s area equally without measuring the central angle.

  Have them use protractors to measure the sector angle, then calculate the fraction of the circle’s area before adding or subtracting. Group members can verify measurements for accuracy.

• During Park Design Challenge, watch for students who ignore curved edges when calculating area, treating them as straight lines.

  Ask them to trace the curved boundary with string, then straighten it to compare lengths. Discuss how curved edges require sector or segment area formulas, not just polygonal approximations.

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Methods used in this brief

• Collaborative Problem-Solving