2D Shapes Formulas

1. Circle

Basic Definitions

Circle: A set of all points equidistant from a fixed center point.

Radius (r): Distance from center to any point on circle
Diameter (d): Distance across circle through center, \(d = 2r\)
Chord: Line segment connecting two points on circle

Circle Formulas

Area: \(A = \pi r^2\)
Circumference: \(C = 2\pi r\) or \(C = \pi d\)
Area of Sector: \(A = \frac{\theta}{360°} \times \pi r^2\)
Length of Arc: \(l = \frac{\theta}{360°} \times 2\pi r\)
Area of Segment: \(A = \text{Area of sector} - \text{Area of triangle}\)

2. Triangle

General Triangle Formulas

Area (base-height): \(A = \frac{1}{2} \times b \times h\)
Perimeter: \(P = a + b + c\)
Semi-perimeter: \(s = \frac{a + b + c}{2}\)
Heron's Formula: \(A = \sqrt{s(s-a)(s-b)(s-c)}\)

Right-Angled Triangle

Pythagoras Theorem: \(c^2 = a^2 + b^2\)
Area: \(A = \frac{1}{2} \times \text{base} \times \text{height}\)
Perimeter: \(P = a + b + c\)

Equilateral Triangle

Area: \(A = \frac{\sqrt{3}}{4}a^2\)
Perimeter: \(P = 3a\)
Height: \(h = \frac{\sqrt{3}}{2}a\)

where \(a\) = side length

Isosceles Triangle

Area: \(A = \frac{b}{4}\sqrt{4a^2 - b^2}\)
Perimeter: \(P = 2a + b\)

where \(a\) = equal sides, \(b\) = base

3. Square

Square Formulas

Area: \(A = a^2\) or \(A = \frac{d^2}{2}\)
Perimeter: \(P = 4a\)
Diagonal: \(d = a\sqrt{2}\)
Side from diagonal: \(a = \frac{d}{\sqrt{2}}\)

where \(a\) = side length, \(d\) = diagonal

4. Rectangle

Rectangle Formulas

Area: \(A = l \times b\)
Perimeter: \(P = 2(l + b)\)
Diagonal: \(d = \sqrt{l^2 + b^2}\)

where \(l\) = length, \(b\) = breadth (width)

5. Parallelogram

Parallelogram Formulas

Area: \(A = b \times h\)
Perimeter: \(P = 2(a + b)\)
Height: \(h = \frac{A}{b}\)

where \(b\) = base, \(h\) = height, \(a\) = side

6. Rhombus

Rhombus Formulas

Area (using diagonals): \(A = \frac{1}{2} \times d_1 \times d_2\)
Area (using base and height): \(A = b \times h\)
Perimeter: \(P = 4a\)
Diagonal relation: \(d_1^2 + d_2^2 = 4a^2\)

where \(d_1, d_2\) = diagonals, \(a\) = side, \(h\) = height

7. Trapezium (Trapezoid)

Trapezium Formulas

Area: \(A = \frac{1}{2}(a + b) \times h\)
Perimeter: \(P = a + b + c + d\)
Median (mid-segment): \(m = \frac{a + b}{2}\)
Area using median: \(A = m \times h\)

where \(a, b\) = parallel sides, \(h\) = height, \(c, d\) = non-parallel sides

8. Kite

Kite Formulas

Area: \(A = \frac{1}{2} \times d_1 \times d_2\)
Perimeter: \(P = 2(a + b)\)

where \(d_1, d_2\) = diagonals, \(a, b\) = pairs of equal sides

9. Regular Polygon

Regular Polygon Formulas

Area: \(A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}\)
Perimeter: \(P = n \times s\)
Interior angle: \(\theta = \frac{(n-2) \times 180°}{n}\)
Exterior angle: \(\alpha = \frac{360°}{n}\)
Sum of interior angles: \(S = (n-2) \times 180°\)

where \(n\) = number of sides, \(s\) = side length

10. Ellipse

Ellipse Formulas

Area: \(A = \pi ab\)
Perimeter (approximate): \(P \approx \pi[3(a+b) - \sqrt{(3a+b)(a+3b)}]\)

where \(a\) = semi-major axis, \(b\) = semi-minor axis

11. Quick Reference Table

Shape	Area Formula	Perimeter Formula
Circle	\(\pi r^2\)	\(2\pi r\)
Square	\(a^2\)	\(4a\)
Rectangle	\(l \times b\)	\(2(l + b)\)
Triangle	\(\frac{1}{2}bh\)	\(a + b + c\)
Equilateral Triangle	\(\frac{\sqrt{3}}{4}a^2\)	\(3a\)
Parallelogram	\(b \times h\)	\(2(a + b)\)
Rhombus	\(\frac{1}{2}d_1 d_2\)	\(4a\)
Trapezium	\(\frac{1}{2}(a+b)h\)	\(a+b+c+d\)
Kite	\(\frac{1}{2}d_1 d_2\)	\(2(a+b)\)

12. Special Area Relationships

Important Area Conversions

Square from diagonal: If diagonal = \(d\), then Area = \(\frac{d^2}{2}\)
Circle from circumference: If \(C = 2\pi r\), then Area = \(\frac{C^2}{4\pi}\)
Rectangle diagonal: \(d^2 = l^2 + b^2\) (Pythagoras)
Equilateral triangle height: \(h = \frac{\sqrt{3}}{2}a\)

Key Points to Remember

Always use consistent units (all cm, all m, etc.)
Area is measured in square units (cm², m², etc.)
Perimeter is measured in linear units (cm, m, etc.)
Use \(\pi \approx 3.14\) or \(\frac{22}{7}\) for calculations
For composite shapes, break them into simpler shapes
Remember: Perimeter is the boundary, Area is the space inside

13. Common Values

Constant	Value	Used For
\(\pi\)	3.14159... or \(\frac{22}{7}\)	Circle calculations
\(\sqrt{2}\)	1.414...	Square diagonal
\(\sqrt{3}\)	1.732...	Equilateral triangle