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Home » Area of Geometric Shapes » Find the Area of the Shaded Region

Find the Area of the Shaded Region

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will use the strategy of calculating the area of a large shape and the area of the smaller shapes it encloses to find the area of the shaded region between them.

Sometimes we are presented with a geometry problem that requires us to find the area of an irregular shape which can't easily be partitioned into simple shapes.

For example, the one in the following drawing, formed by intersecting curves. Here's a good strategy to employ for such problems.

Problem

Two circles, with radii 2 and 1 respectively, are externally tangent (that is, they intersect at exactly one point). An outer circle is tangent to both of these circles. Find the area of the shaded region.

find the area of the shaded region

Strategy

Our usual strategy when presented with complex geometric shapes is to partition them into simpler shapes whose areas are given by formulas we know.

But in this case, and in many similar geometry problems where the shape is formed by intersecting curves rather than straight lines, it is very difficult to do so. For such cases, it is often possible to calculate the area of the desired shape by calculating the area of the outer shape, and then subtracting the areas of the inner shapes.

Tip: When asked to find the "shaded" region, that is often a hint to use this approach - as in another problem we've solved.

In this problem, it is easy to find the area of the two inner circles, since their radii are given. We can also find the area of the outer circle when we realize that its diameter is equal to the sum of the diameters of the two inner circles.

Solution

(1) rsmallInner= 1   //Given
(2) AsmallInner=π*rsmallInner2 = π //(1), Area of a circle
(3) rlargeInner= 2   //Given
(4) AlargeInner=π*rlargeInner2 = 4π //(3), Area of a circle
(5) Douter = 2*(rsmallInner +rlargeInner) //The outer diameter is equal to the sum of the diameters of the two inner circles.
(6) Douter=2*(1+2) //(1), (3), (5)
(7) Router= 3 //(6), the radius is half the diameter
(8) Aouter=π*Router2 = 9π //(7), Area of a circle
(9) Ashaded=Aouter-(AsmallInner+AlargeInner) = 9π-( 4π+π )=4π

« Find the Area of a Circle from 3 Points
Area of a Kite »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected].

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
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    • Tangent Lines
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    • Intersecting Lines and Angles
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  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
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    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
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    • Similar Triangles
    • Triangle Inequalities

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