Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Geometry → Quadrilaterals ↓


Special Quadrilaterals


Introduction to quadrilaterals

In geometry, a quadrilateral is a four-sided polygon with four angles. The word "quadrilateral" is derived from the Latin words "quadri" meaning "four" and "latus" meaning "side." Quadrilaterals come in many different shapes and sizes, and they are a fundamental part of geometric understanding.

Types of quadrilaterals

While all quadrilaterals have four sides, the lengths of the sides and the measures of the angles can vary, creating many different types. Common types of quadrilaterals include squares, rectangles, parallelograms, rhombuses, and trapezoids. These are sometimes known as special quadrilaterals because they each have unique properties.

Social class

A square is a quadrilateral with all sides of equal length and all angles equal to 90 degrees. This means that every square is also a rectangle, a rhombus, and a parallelogram. The formula for the area of a square is given as:

Area = side × side = side 2

Additionally, the perimeter of a square can be calculated as follows:

Perimeter = 4 × side
Side

Rectangle

A rectangle is a quadrilateral with opposite sides equal and all angles equal to 90 degrees. This means that every rectangle is also a parallelogram. The formula for the area of a rectangle is:

Area = length × width

and the perimeter is given by:

Perimeter = 2 × (length + width)
LengthWidth

Quadrilateral

A parallelogram is a quadrilateral whose opposite sides are parallel and equal in length. The opposite angles are also equal. The area of a parallelogram can be found using the formula:

Area = base × height

Meanwhile, the perimeter is given by:

Perimeter = 2 × (base + side)
BaseHeight

Rhombus

A rhombus is a quadrilateral with all sides of equal length, but unlike a square, the angles need not necessarily be 90 degrees. The diagonals of a rhombus bisect each other at right angles. The area of a rhombus can be found using:

Area = (diagonal 1 × diagonal 2 ) / 2

The perimeter is simply 4 times the length of one side:

Perimeter = 4 × side
DiagonalDiagonal

Quadrilateral

A trapezoid (or trapezoid) is a quadrilateral that has at least one pair of parallel sides. These parallel sides are called bases, and the other two sides are called legs. The area can be calculated by the formula:

Area = (base 1 + base 2 ) / 2 × height

The perimeter is the sum of all sides:

Perimeter = base 1 + base 2 + leg 1 + leg 2
Base 1Base 2

Properties of special quadrilaterals

As you can see, each type of special quadrilateral has its own unique properties regarding its sides and angles. Here are some additional properties worth noting:

  • The diagonals of a square are equal and bisect each other at 90 degrees.
  • The diagonals of a rectangle are of equal length and bisect each other, but they are not necessarily at 90 degrees.
  • In a parallelogram the diagonals bisect each other, but they are not necessarily equal.
  • The diagonals of a rhombus are perpendicular to each other.
  • The sum of the interior angles of any quadrilateral is always 360 degrees.

Understanding through examples

Let's look at some examples to apply these formulas and deepen our understanding.

Example 1: Finding the area of a square

Suppose we have a square whose each side is 5 cm. The area of the square can be found as follows:

Area = 5 cm × 5 cm = 25 cm²

Example 2: Determining the perimeter of a rectangle

Consider a rectangle of length 8 cm and width 3 cm. The perimeter is calculated as follows:

Perimeter = 2 × (8 cm + 3 cm) = 2 × 11 cm = 22 cm

Example 3: Finding the area of a parallelogram

If the base of a parallelogram is 10 cm and height 4 cm, then its area will be:

Area = 10 cm × 4 cm = 40 cm²

Example 4: Area of a rhombus

Suppose the lengths of the diagonals of a rhombus are 6 cm and 8 cm. Then, its area can be found as follows:

Area = (6 cm × 8 cm) / 2 = 48 cm² / 2 = 24 cm²

Example 5: Finding the area of a trapezoid

Imagine a trapezoid whose bases are 4 cm and 6 cm and the height is 3 cm. The area is determined as follows:

Area = (4 cm + 6 cm) / 2 × 3 cm = 10 cm / 2 × 3 cm = 15 cm²

Practice problems

Try solving these problems to test your understanding of special quadrilaterals:

  1. What is the area of a rectangle whose length is 12 cm and width is 5 cm?
  2. If the perimeter of a rhombus is 40 cm, what is the length of each side?
  3. Find the area of a square whose diagonal is 8√2 cm.
  4. Find the perimeter of a parallelogram with base 5 cm and side 7 cm.
  5. The lengths of the bases of a trapezium are 10 cm and 14 cm and its height is 6 cm. What is its area?

Conclusion

Special quadrilaterals are an essential part of geometry, each of which has unique properties based on their side lengths and angles. Understanding these properties and how to calculate their area and perimeter can help you not only solve mathematical problems but also understand the larger structure of geometric concepts. Mastering these special quadrilaterals will serve as a strong foundation for further study in geometry.


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Properties of Parallelograms
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Special Quadrilaterals

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