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 -
And narrow hearts like this:
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."
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