Area of the Shapes

What is an Area?

The area is a measure of the amount of space or surface within a two-dimensional shape or object. It is typically measured in square units, such as square inches, square feet, or square meters. The area of a shape or object can be calculated using mathematical formulas, depending on the shape or object's dimensions.

For example,
Area of rectangle = l x b
Where l = Length of the rectangle and B is the breadth of the rectangle
Area of circle = π r²
Where, r = Radius of the circle

Area of the Shapes

The area of a shape refers to the amount of space contained within the boundaries of that shape. Different shapes have different formulas for calculating their area.

Formulas for Area of Shapes

This is divided into two sections 2D and 3D as follows:

Formulas for Area of 2D Shapes

1. Rectangle

Area = Length (l) x Breadth (b)

2. Square

Area = side (a) x side (a) = a²

3. Triangle

Area = (base x height) / 2 = ½ x b x h

4. Circle

Area = π x radius x radius = π r²

5. Parallelogram

Area = Base x height = b x h

6. Trapezium

Area = (Base₁ + Base₂) x height / 2 = ½ h(a + b)

7. Rhombus

Area = (Diagonal 1 x Diagonal 2) / 2 = ½ x d₁ x d₂

8. Kite

Area = (Diagonal 1 x diagonal 2) / 2 = ½ x d₁ x d₂

9. Hexagon

Area = (3 x √3 x (side x side)) / 2 = (3√3)/2 s²

Formulas for Area of 3D Shapes

1. Cube

The area of a cube is 6 times the area of one of its faces, which is a square.
The total surface area of a cube is A = 6a²

Where a is the length of one side of the cube.

2. Sphere

The surface area of a sphere is A = 4πr²

Where r is the radius of the sphere.

3. Cylinder

The formula for the total surface area of a cylinder is A = 2πr² + 2πrh

Where r is the radius of the base and h is the height of the cylinder

Curved surface area (CSA) of cylinder = 2?rh sq. units.

4. Cone

The total surface area of a cone is A = πr² + πrL

Where r is the radius of the base and L is the slant height of the cone.

Curved surface area (CSA) of cone = ?rL sq. units.

5. Rectangular Prism

The total surface area of a rectangular prism is A = 2lw + 2lh + 2wh

Where l is the length, w is the width, and h is the height of the prism

6. Pyramid

The total surface area of a pyramid is A = (B + √[l² + h²])

Where B is the area of the base and l and h are the slant height and height of the pyramid respectively.

7. Triangular Prism

The total surface area of a triangular prism is A = 2bh + 3l

Where b is the base, h is the height, and l is the length of the prism.

Understanding Measurement Units

a. Introduction to Standard Units of Measurement for Area:

• Area is measured in square units. Common standard units for area include square centimeters (cm²), square meters (m²), square inches (in²), and square feet (ft²).
• For example, if we have a square with sides each measuring 5 centimeters, the area would be 5 cm × 5 cm = 25 cm².

b. Converting Between Different Units of Area:

• Students can learn to convert between different units of area using conversion factors.
• For example, to convert 60 cm² to square meters, we divide by 10,000 (since 1 m² = 10,000 cm²):
  60 cm² ÷ 10,000 = 0.006 m²

Real World Problem-Solving with Area

a. Problem-Solving Scenarios Involving Area Calculations in Various Contexts:

Problem 1: A rectangular garden measures 8 meters in length and 5 meters in width. Calculate the area of the garden.
Solution: Area = length × width = 8 m × 5 m = 40 m²

Problem 2: A circular swimming pool has a diameter of 6 meters. Find the area of the pool. (Take π ≈ 3.14)
Solution: Radius (r) = Diameter ÷ 2 = 6 m ÷ 2 = 3 m
Area = π × r² = 3.14 × (3 m)² ≈ 3.14 × 9 m² ≈ 28.26 m²

Problem 3: Emma is making a rectangular painting with a length of 60 cm and a width of 40 cm. She wants to add a border around the painting, which is 5 cm wide. Calculate the total area of the painting and the border.
Solution: Area of painting = length × width = 60 cm × 40 cm = 2400 cm²
Area of border = (60 cm + 2 × 5 cm) × (40 cm + 2 × 5 cm) = 70 cm × 50 cm = 3500 cm²
Total area = Area of painting + Area of border = 2400 cm² + 3500 cm² = 5900 cm²

b. Application of Area Concepts in Everyday Situations:

• Students can apply their understanding of area in real-life scenarios, such as calculating the area of their classroom, playground, or a sports field.
• For example, they can measure the length and width of their classroom in meters and calculate the area to determine how much space is available for seating or activities.