Area is an important geometry and everyday life concept. 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 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.
Shape |
Formula |
Unit |
Square |
Area = side × side |
cm², m², ft² |
Rectangle |
Area = length × breadth |
cm², m², ft² |
Triangle |
Area = ½ × base × height |
cm², m², ft² |
Circle |
Area = π × radius² |
cm², m², ft² |
Parallelogram |
Area = base × height |
cm², m², ft² |
Rhombus |
Area = ½ × diagonal₁ × diagonal₂ |
cm², m², ft² |
Trapezium |
Area = ½ × (a + b) × height |
cm², m², ft² |
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
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
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
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
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
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
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
Shape |
Surface Area Formula |
Unit |
Cube |
Surface Area = 6 × side² |
cm², m² |
Cuboid |
Surface Area = 2(lb + bh + hl) |
cm², m² |
Cylinder |
Surface Area = 2πr(h + r) |
cm², m² |
Cone |
Surface Area = πr(l + r) |
cm², m² |
Sphere |
Surface Area = 4πr² |
cm², m² |
Hemisphere |
Surface Area = 3πr² |
cm², m² |
Pyramid |
Surface Area = Base Area + ½ × Perimeter × Slant Height |
cm², m² |
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
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
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
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
Formula:
Surface Area = 4πr²
Example:
Find surface area of a sphere with radius 6 cm
Steps:
r² = 36
Surface Area = 4 × 3.14 × 36 = 452.16 cm²
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
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 memorizing formulas, know the steps with reason, and then go on to practice applying it in the real world. With repetition, the area is not only simple-but instinctual.
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:
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
Divide the quadrilateral into two triangles.
Use the triangle area formula for each part.
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.
Explore more essential math concepts with Orchids International School! Uncover the power of real-world applications in your learning journey.