What is Area: Definition and 1000 Discussions

Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.
Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

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1. Find area of the shaded part in the given diagram

In my working, i have the following approach; Using area of semi -circle and area of sector concept; ##x +z = \dfrac{9π}{2}## ##x +y = \dfrac{9π}{2}## ##z+p = \dfrac{9π}{2}## On solving the simultaneous equations, ##⇒x=p## then, ##x=\dfrac {9π}{4} - \left(\dfrac{1}{2} ×3 ×3\right) =...

45. Solving this problem that involves the area of triangle

I am still looking at this question. One thing that i know is that the distance ##AB=\dfrac {(λ+1)\sqrt {2λ^2-2λ+1}}{λ^2}## distance ##OA=\sqrt 2## distance ##OB=\dfrac{\sqrt{λ^2+1}}{λ^2}## Perpendicular distance from point ##B## to the line ##OA=\dfrac{\sqrt{2(λ^4+2λ^3+2)}}{2λ^2}## Therefore...
46. I Calculating the surface area of a sphere using dA

Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...
47. I Polygon Coordinates given the Area and Center point

I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following - Center point is known - area is known - Point A is known - Points B, C, and D are UNKNOWN I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...
48. Finding the area of a double integral using dxdy instead of dydx

I have the solution for this problem using dydx as the area. Worse yet, I cannot find another solution for it. Everyone seems to just magically pick dydx without thinking and naturally this is frustrating as learning the correct choice is 99.9% of the battle... So, I was curious how one might...
49. Find the area of the shaded region in the inscribed circle on square

Find the solution here; Find my approach below; In my working i have; ##A_{minor sector}##=##\frac {128.1^0}{360^0}×π×5×5=27.947cm^2## ##A_{triangle}##=##\frac {1}{2}####×5×5×sin 128.1^0=9.8366cm^2## ##A_3##=##\frac {90^0}{360^0}####×π×10×10##=##78.53cm^2## ##A_{major...
50. Finding the Area of a Sector using Trigonometric Functions

My interest is on part (c) only. Wow, this was a nice one! boggled me a little bit anyways; my last steps to solution, ##A= \dfrac{1}{2} ×16.8319^2 × 0.5707=80.84cm^2## bingo! Any other approach apart from using sine? Cheers guys