Area – How Much Space a Shape Covers

Area measures how much flat space a shape covers. It is one of the most practical measurements in everyday life — from tiling a floor and painting a wall to calculating the size of a solar panel or a sports pitch.

Some of the oldest surviving area problems appear in the Rhind Papyrus, an Egyptian mathematical text dating to around 1650 BCE, which shows scribes working out the area of circular and triangular fields for tax and land-division purposes. For over 2,000 years, mathematicians also chased an especially stubborn area puzzle known as “squaring the circle” — trying to construct, using only a compass and straightedge, a square with the exact same area as a given circle. It was not until 1882 that the German mathematician Ferdinand von Lindemann finally proved this to be impossible, by showing that π is a transcendental number. Area calculations remain central to everyday economics too: real estate is priced by the square foot or square metre, precisely because area is the fairest way to compare how much usable space different properties actually offer.

What Is Area?

Area is measured in square units: cm², m², km². It tells you how many unit squares fit inside a shape. Area is always a positive number.

Area Formulas for Common Shapes

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 = perpendicular height
Parallelogram	A = b × h	b = base, h = perpendicular height
Trapezium	A = ½(a + b) × h	a, b = parallel sides, h = height
Circle	A = πr²	r = radius
Sector	A = (θ/360) × πr²	θ = angle in degrees
Kite or Rhombus	A = ½ × d₁ × d₂	d₁, d₂ = diagonals

Worked Examples

Find the area of a triangle with base 12 cm and height 7 cm.

A = ½ × 12 × 7 = 42 cm².

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

A = ½(5 + 9) × 4 = ½ × 14 × 4 = 28 cm².

Find the area of a circle with diameter 14 cm.

r = 7 cm. A = π × 49 = 49π ≈ 153.94 cm².

Area of Compound Shapes

Split the shape into simpler known shapes, calculate each area, then add (or subtract) them.

A shape is a rectangle 10 × 6 with a semicircle of diameter 6 on top. Find the total area.

Rectangle: 10 × 6 = 60. Semicircle: ½π(3)² = 4.5π ≈ 14.14. Total: ≈ 74.14 cm².

Key Takeaways

Area is always in square units (cm², m², etc.).
Triangle: ½bh. Rectangle: lw. Circle: πr². Trapezium: ½(a+b)h.
For compound shapes: split, calculate each part, add or subtract.
Always use the perpendicular height, not the slant side.

Practice: Area Calculations

Find the Area