Area of Shapes

Introduction

Learning to calculate area is not only important in geometry but is equally important in everyday life. It informs us of the amount of flat space a shape takes up. Whether you're tiling, painting, or planning plots, knowing how to work out an area makes you a wiser math and decision-maker. Throughout this guide, you'll find the formulas and in-depth examples of working out areas for 2D and 3D shapes.

 

Table of Contents

• Area Definition
• 2D Shapes - Area Formulae Table
• 2D Shapes and Area Formulas with Solved Examples
• 3D Shapes - Surface Area Formulae Table
• 3D Shapes and Surface Area Formulas with Solved Examples
• Conclusion
• FAQs on Area of Shapes

 

Area Definition

Area is the quantity of surface covered by a two-dimensional shape or figure. It is always in terms of square units (such as cm², m², or ft²), one unit by one unit coverage.

Example of area: If a square has a side length of 1 cm, its area is 1 cm × 1 cm = 1 cm².

Unlike perimeter (measuring length around a shape), area calculates the space inside the shape.

 

2D Shapes - Area Formulae Table

2D Shapes and Area Formulas with Solved Examples

Square

Formula:

Area = side × side

Example:

Calculate the area of a square whose side is 6 cm.

Steps:

Identify the side = 6 cm

Use formula: Area = 6 × 6 = 36 cm²

 

Rectangle

Formula:

Area = length × breadth

 

Example:

Find the area of a rectangle which is 8 cm long and 5 cm broad.

 

Steps:

Length = 8 cm, Breadth = 5 cm

Area = 8 × 5 = 40 cm²

 

Triangle

Formula:

Area = (1/2) × base × height

 

Example:

Find the area of a triangle whose base is 10 cm and height is 6 cm.

 

Steps:

Base = 10 cm, Height = 6 cm

Area = (1/2) × 10 × 6 = (1/2) × 60 = 30 cm²

 

Circle

Formula:

Area = π × radius²

 

Example:

Calculate the area of a circle with a radius of 7 cm (Use π = 3.14)

 

Steps:

Radius = 7 cm

Area = 3.14 × 7 × 7 = 3.14 × 49 = 153.86 cm²

 

Parallelogram

Formula:

Area = base × height

 

Example:

Calculate the area of a parallelogram with a base of 12 cm and height of 5 cm.

 

Steps:

Base = 12 cm, Height = 5 cm

Area = 12 × 5 = 60 cm²

 

Rhombus

Formula:

Area = (1/2) × (diagonal₁ × diagonal₂)

Example:

If diagonals of a rhombus are 10 cm and 8 cm, find its area.

 

Steps:

Diagonal₁ = 10 cm, Diagonal₂ = 8 cm

Area = (1/2) × 10 × 8 = (1/2) × 80 = 40 cm²

 

Trapezium

Formula:

Area = (1/2) × (sum of parallel sides) × height

Example:

Find the area of trapezium with parallel sides of length 9 cm and 5 cm, and height 4 cm.

Steps:

Parallel sides = 9 cm and 5 cm

Height = 4 cm

Area = (1/2) × (9 + 5) × 4 = (1/2) × 14 × 4 = 7 × 4 = 28 cm²

 

3D Shapes - Surface Area Formulae Table

 

3D Shapes and Surface Area Formulas with Solved Examples

Cube

Formula:

Surface Area = 6 × side²

 

Example:

Find the surface area of a cube whose side is 3 cm.

Steps:

Side = 3 cm

Surface Area = 6 × 3 × 3 = 6 × 9 = 54 cm²

 

Cuboid

Formula:

Surface Area = 2(lb + bh + hl)

 

Example:

Find surface area of cuboid whose l = 4 cm, b = 3 cm, h = 2 cm

Steps:

lb = 4×3 = 12, bh = 3×2 = 6, hl = 2×4 = 8

Sum = 12 + 6 + 8 = 26

Surface Area = 2 × 26 = 52 cm²

 

Cylinder

Formula:

Surface Area = 2πr(h + r)

 

Example:

Find surface area of a cylinder with r = 5 cm, h = 10 cm

Steps:

h + r = 10 + 5 = 15

Surface Area = 2 × 3.14 × 5 × 15 = 471 cm²

Answer: 471 cm²

 

Cone

Formula:

Surface Area = πr(l + r), where l = slant height

 

Example:

Find surface area of a cone with r = 3 cm, l = 5 cm

Steps:

l + r = 5 + 3 = 8

Surface Area = 3.14 × 3 × 8 = 75.36 cm²

Answer: 75.36 cm²

 

Sphere

Formula:

Surface Area = 4πr²

 

Example:

Find surface area of a sphere with a radius of 6 cm.

Steps:

r² = 36

Surface Area = 4 × 3.14 × 36 = 452.16 cm²

 

Hemisphere

Formula:

Surface Area = 3πr²

 

Example:

Find surface area of hemisphere whose radius is 4 cm

Steps:

r² = 16

Surface Area = 3 × 3.14 × 16 = 150.72 cm²

 

Conclusion

By learning how to measure the area of 2D and 3D objects, you're giving yourself useful math skills that translate into such professions as engineering, architecture, and everyday budgeting. Begin with memorising formulas, know the steps with reason, and then go on to practise applying it in the real world. With repetition, the area is not only simple-but instinctual.

 

Frequently Asked Questions on Area of Shapes

1. How do you find the area of shapes?

Ans: To find the area of a shape, use a specific formula that matches its geometry. Here's how:

• Square: Area = side × side

• Rectangle: Area = length × breadth

• Triangle: Area = ½ × base × height

• Circle: Area = π × radius²

• Parallelogram: Area = base × height

• Trapezium: Area = ½ × (sum of parallel sides) × height

• Irregular shapes: Break the shape into smaller regular shapes, find their areas, then add them up
  
  

2. What are the 7 most common surface areas?

Ans: Here are 7 commonly used surface area formulas in geometry and real-world applications:

• Cube: Surface Area = 6 × side²

• Cuboid: Surface Area = 2(lb + bh + hl)

• Cylinder: Surface Area = 2πr(h + r)

• Cone: Surface Area = πr(l + r)

• Sphere: Surface Area = 4πr²

• Hemisphere: Surface Area = 3πr²

• Pyramid: Surface Area = Base Area + ½ × Perimeter × Slant Height

Each shape’s surface area measures the total outer surface it occupies.

 

3. How to find an area of 4 unequal sides?

Ans: If a shape has 4 unequal sides, it's likely a quadrilateral, and the formula depends on the type:

For a general quadrilateral, use Bretschneider’s formula (when all sides and one diagonal or angle are known):

Area = √[(s - a)(s - b)(s - c)(s - d) - abcd × cos²(θ/2)]

Where:

• a, b, c, d = sides

• s = semi-perimeter = (a + b + c + d) / 2

• θ = angle between two opposite sides

Shortcut (if shape is irregular but closed):

1. Divide the quadrilateral into two triangles.

2. Use the triangle area formula for each part.

3. Add both areas.

 

4. How to find the area of an irregular shape?

Ans: To find the area of an irregular shape:

• Step 1: Break the shape into known shapes (like triangles, rectangles, or circles).

• Step 2: Use the appropriate formula for each smaller shape.

• Step 3: Add all individual areas.

Example:
If an L-shaped plot consists of two rectangles:

• Rectangle 1: 6 m × 4 m = 24 m²

• Rectangle 2: 3 m × 2 m = 6 m²

• Total Area = 24 + 6 = 30 m²

In some cases (like plotted land), you can also apply the grid method or use coordinate geometry if vertex points are known.

 

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