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.

• Class 10
• Mathematics
• 4 activities
• 35 min–50 min

Learning Objectives

1. Calculate the area of composite figures formed by combining circles, sectors, and polygons.
2. Analyze shaded regions within combined figures to determine whether areas should be added or subtracted.
3. Compare different strategies for finding the area of complex shapes and justify the most efficient method.
4. Design a step-by-step approach to solve problems involving areas of combined plane figures.

Project-Based Learning

⏱ 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.

Prepare & details

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.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Project-Based Learning

⏱ 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.

Prepare & details

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.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Project-Based Learning

⏱ 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.

Prepare & details

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.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Project-Based Learning

⏱ 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.

Prepare & details

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.

Setup: Standard classroom of 40–50 students; printed task and role cards are recommended over digital display to allow simultaneous group work without device dependency.

Materials: Printed driving question and role cards, Chart paper and markers for group outputs, NCERT textbooks and supplementary board materials as base resources, Local data sources — newspapers, community interviews, government census data, Internal assessment rubric aligned to board project guidelines

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Quick Check

After Paper Cutouts, present a diagram of a square with a semicircle attached to one side. Ask students to write down the formulas for the square’s area, semicircle’s area, and total area, and circle which shapes need to be added or subtracted.

Exit Ticket

During Shaded Relay, provide a complex shaded region, such as a rectangle with a quarter-circle cut from one corner. Ask students to calculate the area and explain in one sentence how they handled the curved part in their calculation.

Discussion Prompt

After GeoGebra Exploration, show two different methods for calculating the area of a figure with overlapping circular regions. Ask students to discuss in pairs which method is more efficient and why, focusing on the steps involving subtraction of shared areas.

Extensions & Scaffolding

• After completing GeoGebra Exploration, challenge students to design a logo with at least four combined shapes and calculate its total area, including shaded regions.
• For students who struggle, provide pre-cut shapes with labelled dimensions and ask them to arrange and compute one figure at a time before combining.
• During Park Design Challenge, invite students to research real-world applications of such designs, like garden layouts or playground structures, and present their findings to the class.

Key Vocabulary

Composite Figure	A shape made up of two or more basic geometric shapes, such as circles, sectors, rectangles, or triangles.
Area of Sector	The portion of a circle enclosed by two radii and an arc, calculated using a fraction of the circle's total area.
Shaded Region	A specific part of a composite figure, often irregular, whose area needs to be calculated by combining or subtracting the areas of simpler shapes.
Mensuration	The branch of geometry concerned with the measurement of length, area, and volume of figures.