Area Calculation: Comprehensive Guide
What is Area?
Area is the amount of space inside the boundary of a 2-dimensional shape. It is measured in square units, such as square centimeters (cm²), square meters (m²), or square inches (in²).
Common Area Formulas
| Shape | Formula | Variables |
|---|---|---|
| Square | A = s² | s = side length |
| Rectangle | A = l × w | l = length, w = width |
| Triangle | A = ½ × b × h | b = base, h = height |
| Circle | A = π × r² | r = radius |
Area of a Square
Formula:
Area of a square = s × s = s²
Where s is the length of one side of the square.
Ways to Calculate Square Area:
- Using the side length: A = s²
- Using the diagonal: A = d² ÷ 2 (where d is the diagonal length)
- Using the perimeter: A = (P ÷ 4)² (where P is the perimeter)
Example 1: Basic Square Area
Find the area of a square with side length 5 cm.
Solution:
Given: s = 5 cm
Area = s² = 5² = 5 × 5 = 25 cm²
Example 2: Square Area from Diagonal
Find the area of a square with a diagonal of 8 inches.
Solution:
Given: Diagonal d = 8 inches
First, find the side length using the Pythagorean relationship: d = s√2
s = d ÷ √2 = 8 ÷ 1.414 = 5.657 inches
Area = s² = 5.657² = 32 in²
Alternative direct formula: Area = d² ÷ 2 = 8² ÷ 2 = 64 ÷ 2 = 32 in²
Example 3: Square Area from Perimeter
Find the area of a square with a perimeter of 20 meters.
Solution:
Given: Perimeter P = 20 meters
Side length s = P ÷ 4 = 20 ÷ 4 = 5 meters
Area = s² = 5² = 25 m²
Area of a Rectangle
Formula:
Area of a rectangle = length × width = l × w
Ways to Calculate Rectangle Area:
- Using length and width: A = l × w
- Using diagonal and one side: A = w × √(d² - w²) (where d is the diagonal)
- Using perimeter and aspect ratio: If perimeter = 2(l + w) and l:w ratio is known
Example 1: Basic Rectangle Area
Find the area of a rectangle with length 8 m and width 5 m.
Solution:
Given: l = 8 m, w = 5 m
Area = l × w = 8 × 5 = 40 m²
Example 2: Rectangle Area from Perimeter and Ratio
Find the area of a rectangle with perimeter 30 cm and length to width ratio of 2:1.
Solution:
Given: Perimeter = 30 cm, l:w = 2:1
Let's say w = x, then l = 2x (from the ratio)
Perimeter = 2(l + w) = 2(2x + x) = 2(3x) = 6x
30 = 6x
x = 5
Therefore, w = 5 cm and l = 2 × 5 = 10 cm
Area = l × w = 10 × 5 = 50 cm²
Example 3: Rectangle Area from Diagonal and Width
Find the area of a rectangle with diagonal 13 cm and width 5 cm.
Solution:
Given: Diagonal d = 13 cm, w = 5 cm
Using the Pythagorean theorem: d² = l² + w²
13² = l² + 5²
169 = l² + 25
l² = 144
l = 12 cm
Area = l × w = 12 × 5 = 60 cm²
Area of a Triangle
Basic Formula:
Area of a triangle = ½ × base × height = ½ × b × h
Ways to Calculate Triangle Area:
- Using base and height: A = ½ × b × h
- Using side lengths (Heron's formula): A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2
- Using coordinates (Shoelace formula): A = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
- Using two sides and included angle: A = ½ × a × b × sin(C)
- Using radius of circumscribed circle: A = (abc)/(4R) where R is the radius
Example 1: Basic Triangle Area
Find the area of a triangle with base 8 cm and height 5 cm.
Solution:
Given: b = 8 cm, h = 5 cm
Area = ½ × b × h = ½ × 8 × 5 = 20 cm²
Example 2: Triangle Area using Heron's Formula
Find the area of a triangle with sides 5 cm, 6 cm, and 7 cm.
Solution:
Given: a = 5 cm, b = 6 cm, c = 7 cm
Using Heron's formula:
s = (a + b + c) ÷ 2 = (5 + 6 + 7) ÷ 2 = 18 ÷ 2 = 9
Area = √(s(s-a)(s-b)(s-c))
Area = √(9(9-5)(9-6)(9-7))
Area = √(9 × 4 × 3 × 2)
Area = √216 = 14.7 cm²
Example 3: Triangle Area using Two Sides and Angle
Find the area of a triangle with two sides of 4 in and 6 in, and the included angle of 30°.
Solution:
Given: a = 4 in, b = 6 in, C = 30°
Area = ½ × a × b × sin(C)
Area = ½ × 4 × 6 × sin(30°)
Area = ½ × 4 × 6 × 0.5
Area = 6 in²
Special Triangle Types:
- Equilateral triangle: A = (√3 ÷ 4) × s²
- Right triangle: A = ½ × (product of legs)
- Isosceles triangle: A = ½ × b × √(a² - (b²÷4))
Area of a Circle
Formula:
Area of a circle = π × r² = π × (d ÷ 2)²
Where r is the radius and d is the diameter.
The value of π (pi) is approximately 3.14159 or 22/7 for calculations.
Ways to Calculate Circle Area:
- Using radius: A = πr²
- Using diameter: A = π(d/2)² = πd²/4
- Using circumference: A = C²/(4π) where C is the circumference
Example 1: Basic Circle Area
Find the area of a circle with radius 5 cm.
Solution:
Given: r = 5 cm
Area = πr² = π × 5² = π × 25 = 78.54 cm²
Example 2: Circle Area from Diameter
Find the area of a circle with diameter 12 in.
Solution:
Given: d = 12 in
Radius r = d ÷ 2 = 12 ÷ 2 = 6 in
Area = πr² = π × 6² = π × 36 = 113.1 in²
Alternative: Area = πd²/4 = π × 12²/4 = π × 144/4 = 113.1 in²
Example 3: Circle Area from Circumference
Find the area of a circle with circumference 31.4 cm.
Solution:
Given: Circumference C = 31.4 cm
We know C = 2πr
r = C ÷ (2π) = 31.4 ÷ (2π) = 31.4 ÷ 6.28 = 5 cm
Area = πr² = π × 5² = π × 25 = 78.54 cm²
Alternative: Area = C²/(4π) = 31.4²/(4π) = 985.96/12.56 = 78.5 cm²
Example 4: Area of Sector and Segment
Find the area of a sector with radius 10 cm and central angle 60°.
Solution:
Given: r = 10 cm, θ = 60° = π/3 radians
Area of sector = (θ/2) × r² = (π/3)/2 × 10² = π/6 × 100 = 52.36 cm²
Area of segment = Area of sector - Area of triangle
Area of triangle = (1/2) × r² × sin(θ) = (1/2) × 10² × sin(60°) = 50 × 0.866 = 43.3 cm²
Area of segment = 52.36 - 43.3 = 9.06 cm²
Practice Quiz on Area Calculations
Test your knowledge with these practice questions:
1. What is the area of a square with side length 7 meters?
2. Find the area of a rectangle with length 12 cm and width 5 cm.
3. Calculate the area of a triangle with base 8 inches and height 6 inches.
4. What is the area of a circle with radius 4 cm? (Use π = 3.14)
5. A square has a perimeter of 36 cm. What is its area?
6. Find the area of a triangle with sides 5 cm, 12 cm, and 13 cm.
7. The diameter of a circle is 14 feet. What is its area? (Use π = 3.14)
8. What is the area of a rectangle with length 9 m and diagonal 15 m?
Area Calculator
Use the calculators below to find the area of different shapes: