Area (geometry)
Area is the measure of the two-dimensional region enclosed within a closed geometric figure, expressed in square units. It quantifies how much flat surface a shape covers and is fundamental in fields ranging from architecture and land surveying to physics and engineering. Different shapes have distinct area formulas derived from their geometric properties, such as A = πr² for a circle or A = ½bh for a triangle.
Key Formula
A_rectangle = l × w; A_circle = π × r²; A_triangle = (1/2) × b × h
LaTeX: A_{\text{rectangle}} = l \times w, \quad A_{\text{circle}} = \pi r^2, \quad A_{\triangle} = \tfrac{1}{2}bh
| Symbol | Meaning | Unit |
|---|---|---|
| l | Length of rectangle | m |
| w | Width of rectangle | m |
| r | Radius of circle | m |
| b | Base of triangle | m |
| h | Height of triangle | m |
Worked Example
Problem
A room has a rectangular floor 8 m long and 5 m wide, with a semicircular alcove of radius 2 m cut from one short wall. Find the net floor area.
Solution
Step 1 — Rectangle area: A_rect = 8 × 5 = 40 m². Step 2 — Semicircle area: A_semi = πr²/2 = 3.14159 × 4/2 ≈ 6.28 m². Step 3 — Net area: A_net = 40 − 6.28 = 33.72 m².
Answer
Net floor area ≈ 33.72 m²
Area Formulas for Common 2D Shapes
| Shape | Formula | Variables | Example Value |
|---|---|---|---|
| Square | A = s² | s = side length | s = 6 cm → A = 36 cm² |
| Rectangle | A = l × w | l = length, w = width | l=8, w=5 → A = 40 m² |
| Triangle | A = ½bh | b = base, h = height | b=10, h=6 → A = 30 cm² |
| Circle | A = πr² | r = radius | r = 7 → A ≈ 153.94 cm² |
| Trapezium | A = ½(a+b)h | a,b = parallel sides, h = height | a=4,b=6,h=5 → A=25 cm² |
| Parallelogram | A = bh | b = base, h = perpendicular height | b=9, h=4 → A = 36 cm² |
Interactive Tools
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Related Terms
Surface Area
Surface area is the total area of all the outer faces or surfaces of a three-dimensional solid, expressed in square units. It measures how much material is needed to cover an object completely and is critical in applications such as packaging design, heat transfer calculations, and chemical reaction rates (which depend on exposed surface area). For a closed solid, the surface area is found by summing the areas of every face or, for curved surfaces, by integration.
Volume (geometry)
Volume is the measure of the three-dimensional space enclosed by a closed surface, expressed in cubic units. It quantifies the capacity or amount of space a solid object occupies, and is essential in engineering, physics, chemistry, and everyday applications like packaging and fluid storage. Each solid shape has a specific volume formula derived from its geometry, such as V = lwh for a cuboid or V = (4/3)πr³ for a sphere.
Radius
The radius of a circle or sphere is the distance from the centre to any point on its boundary. It is one of the most fundamental measurements in circular geometry, directly determining the size of the circle. The radius relates to the diameter by the equation r = d/2 and appears in formulas for circumference, area, and arc length.
From Latin "area" meaning "a vacant piece of level ground" or "an open space". The mathematical use evolved from the Roman agrimensores (land surveyors), with rigorous definitions appearing in Euclid's "Elements" around 300 BCE.