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Home » Area of Geometric Shapes » Area of a Heart Shape

Area of a Heart Shape

Last updated: Jul 30, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

When studying the topic of "area", a geometry question that comes up frequently is how to find the area of a non-standard shape. In this post, we'll solve a common example - finding the area of a heart shape.

A "heart" is not a well defined description. We have wide hearts like this -

Wide heart illustration

And narrow hearts like this:

Narrow heart illustration

So when solving this problem, unless we have some specific measurements or instructions that say otherwise, we will need to make some simplifying assumptions.

Problem

In the shape below, D is the length between its bottom tip and the bottom of the indent between the rounded areas. Find the area of the "heart shape."

An illustration of square with semicircles in blue.

Strategy

In this case, we will assume that the area of a "heart" shape is made up of a square with two semicircles attached to its sides. then, we will use the strategy of breaking down a complex shape into the simple shapes that make up its parts.

With that assumption, we have a square, whose area is given by the formula Asquare=a2, and two semicircles. The distance D is simply the square's diagonal.

The area of each semicircle is given by the formula Asemicircle=π*r2/2. And in our case, the diameter is simply the side of the square, a, so r=a/2

To find the length of the side, a, from the diagonal, D, we divide by √2, as we have shown here - How to find the side of a square from its diagonal.

So, the area of the square is a2= (D/√2)2=D2/2. The area of each semicircle is π*r2/2= π*(D/2√2)2/2, and we have two of them, so together, the area of the two semicircles is π*(D/2√2)2=π*D2/8

And the area of the heart shape is Asquare+2Asemicircle = D2/2+π*D2/8=(1+π/4)D2/2

« Prove that the Tangent is Perpendicular to the Radius
A Parallelogram with Equal Diagonals is a Rectangle »

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…

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