Mathematics · Class 8 · Mensuration and Surface Analysis · Term 2

Area of Rhombus and General Quadrilaterals

Students will derive and apply formulas for the area of a rhombus and general quadrilaterals.

• CBSE Learning Outcomes
• CBSE: Mensuration - Area of Polygons - Class 8

About This Topic

Surface area of solids moves the concept of area into the third dimension. Students learn to calculate the Total Surface Area (TSA) and Lateral Surface Area (LSA) of cuboids, cubes, and cylinders. The distinction between TSA and LSA is vital, LSA covers only the 'walls', while TSA includes the 'floor' and 'ceiling'. This has immediate practical applications, such as calculating the amount of paint needed for a room or the metal sheet required for a storage tank.

The CBSE curriculum focuses on understanding the 'net' of these solids. For example, a cylinder's lateral surface is actually a rectangle when 'unrolled'. This visual connection is the key to understanding the formulas. This topic comes alive when students can physically model the patterns by unrolling cardboard cylinders or unfolding boxes to see the 2D shapes that make up the 3D surface.

Key Questions

1. Justify why the area of a rhombus is half the product of its diagonals.
2. Explain how to find the area of an irregular quadrilateral by dividing it into triangles.
3. Compare the area formula of a rhombus with that of a parallelogram.

Learning Objectives

• Derive the formula for the area of a rhombus using its diagonals.
• Calculate the area of a rhombus given the lengths of its diagonals.
• Explain the method for calculating the area of an irregular quadrilateral by decomposition into triangles.
• Compare the area formula of a rhombus with that of a parallelogram.
• Apply formulas to find the area of general quadrilaterals in practical contexts.

Before You Start

Area of Parallelograms

Why: Students need to understand the base times height formula for parallelograms to compare it with the rhombus area formula.

Area of Triangles

Why: The method for finding the area of irregular quadrilaterals relies on decomposing them into triangles, so prior knowledge of triangle area is essential.

Properties of Quadrilaterals

Why: Understanding the specific properties of a rhombus, such as its diagonals, is foundational for deriving its area formula.

Key Vocabulary

Rhombus	A quadrilateral with all four sides equal in length. Its diagonals bisect each other at right angles.
Diagonals of a Rhombus	Line segments connecting opposite vertices of a rhombus. They are perpendicular bisectors of each other.
Quadrilateral	A polygon with four sides and four vertices.
Decomposition	The process of breaking down a complex shape into simpler shapes, such as triangles, to make calculations easier.
Area	The measure of the two-dimensional space enclosed by a shape.

Watch Out for These Misconceptions

Common Misconception

Confusing the radius and diameter when using the cylinder formula.

What to Teach Instead

In the 'Unrolling Cylinder' activity, have students physically measure both. When they see that the circumference (2*pi*r) matches the rectangle's width, the role of the radius becomes clear through direct measurement.

Common Misconception

Forgetting to include all six faces in the TSA of a cuboid.

What to Teach Instead

Use a 'Station Rotation' where students must label each face of a box (Top, Bottom, Front, Back, Left, Right) before calculating. Physically touching each face ensures they don't miss any in their sum.

Active Learning Ideas

Inquiry Circle: The Unrolling Cylinder

Groups take a cardboard cylinder (like a kitchen roll tube), measure its height and diameter, and then cut it vertically to 'unroll' it. They discover that the width of the resulting rectangle is the circumference of the circle.

30 min·Small Groups

Stations Rotation: The Packaging Challenge

Stations have different objects: a matchbox, a cylindrical tin, and a cube. Students must calculate the minimum amount of wrapping paper (TSA) needed for each, accounting for all faces.

40 min·Small Groups

Think-Pair-Share: LSA vs TSA

The teacher presents a scenario: 'You are painting the walls and ceiling of a room but not the floor.' Students think about which parts of the cuboid formula to use, pair up to compare, and share their 'modified' formula.

20 min·Pairs

Real-World Connections

• Architects and civil engineers use calculations involving areas of quadrilaterals, including rhombuses, when designing floor plans for buildings or planning the layout of parks and public spaces.
• Kite makers often use the properties of rhombuses and other quadrilaterals to design kites with specific aerodynamic qualities, ensuring stability and lift.
• Tiling and flooring professionals calculate the area of rooms and the shapes of tiles, which can sometimes be rhomboidal, to determine the quantity of material needed and to plan the pattern.

Assessment Ideas

Quick Check

Present students with a diagram of a rhombus and provide the lengths of its diagonals. Ask them to calculate the area using the formula and show their steps. Then, pose a question: 'If you were to cut this rhombus along one diagonal, what two shapes would you get and what is the area of each?'

Discussion Prompt

Show students an irregular quadrilateral. Ask: 'How can we find the area of this shape if we don't have a direct formula for it?' Guide them to suggest dividing it into triangles. Then ask: 'What information would we need about these triangles to calculate the total area?'

Exit Ticket

Give students two problems: 1. Calculate the area of a rhombus with diagonals 10 cm and 12 cm. 2. A quadrilateral is divided into two triangles with areas 25 sq cm and 30 sq cm. What is the total area of the quadrilateral? Students write their answers and one sentence explaining the strategy used for each.

Frequently Asked Questions

What is the difference between Lateral and Total Surface Area?

Lateral Surface Area (LSA) is the area of the side faces only, excluding the top and bottom bases. Total Surface Area (TSA) includes all faces of the solid. For a room, LSA would be the area of the four walls, while TSA would include the floor and ceiling.

How is the LSA of a cylinder derived?

If you 'unroll' the curved surface of a cylinder, it forms a rectangle. The height of the rectangle is the height of the cylinder (h), and the width is the circumference of the circular base (2*pi*r). Thus, the area is 2*pi*r*h.

What is the formula for the TSA of a cuboid?

The TSA of a cuboid is 2(lb + bh + hl), where l is length, b is breadth, and h is height. This formula adds the areas of the three pairs of identical opposite faces.

How can active learning help students understand surface area?

Active learning, such as 'unrolling' a cylinder or unfolding a box, turns a 3D formula into a 2D reality. When students see a cylinder become a rectangle, the formula 2*pi*r*h makes immediate sense. This hands-on approach prevents the common mistake of 'formula mixing' because students can mentally 'unfold' the shape to verify their calculations. It builds a deeper, more permanent understanding of how 3D shapes are constructed.