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We are building a heart-shaped pond, and I want to find out how much water it will hold, without actually filling it yet. How would I do this?
It's a little complicated, so bring out that pencil and ruler! A compass or other circle drawing tool would be very helpful too. Remember that a heart is basically two semicircles on top of a triangle.
Draw two congruent circles tangent to each other, side by side. This means that the two circles have exactly one point touching each other. We'll call that point point "T". Draw a perfectly horizontal line through point T so that it divides both circles into semicircles. Label the points where the line touches the circles "E" and "F" It should look like the two diameters of the circles combining to make one big line, EF.
From point T again, draw a line straight down, perpendicular to line EF to the desired length. End the line at point G. Form a triangle with points EFG. You should now have what looks like a heart with the circles overlapping the triangle bottom.
Now to find the area, first find the area of the two semicircles. The area of a circle = πxr². Notice how lines ET and TF are the diameters. They are also equal. So the radius should equal 1/2 of either diameter. Since the diameters are equal, the radius is also equal to 1/4 of the whole line EF. You should have the measurement for either of those lines, so divide by the corresponding number and get r then plug it into the area formula. You get the area of one of the small circles. It is also the area of two semicircles. We do not multiply this area by two, because we only need the area of two semicircles (the two "humps" of the heart). Write the area down somewhere where you won't forget it.
Now we find the area of the triangle using the formula 1/2 x base x height. The height of the triangle is TG since it's perpendicular and the base of the triangle is EF. Plug in these values and find the area of the bottom triangle. Then add the two areas together to get the area of your heart. Multiply this by the depth of your pond and you get your volume.
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