﻿ Area of the Shapes (2D & 3D Shapes) - CREST Olympiads

# Area of the Shapes

## Area of the Shapes - Sub Topics

• What is an Area?
• Area of the Shapes
• Formulas for Area of Shapes
• Understanding Measurement Units
• Real-World Problem-Solving with Area
• ## 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:

### 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

### 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.

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