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8.8 Volumes with Cross Sections: Triangles and Semicircles

1 min readβ€’june 8, 2020

Anusha Tekumulla

Anusha Tekumulla


AP Calculus AB/BC ♾️

279Β resources
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Using Triangles as Cross Sections

This topic is very similar to topic 8.7 except now we’re using triangles and semicircles as cross sections. To find the area of a shape using a triangle or semicircle cross section, take a look at the example below.Β 

πŸ” Example Problem: Finding the Volumes of a Shape with Triangle Cross SectionsΒ Β 

Let’s say we are asked to find the volume of a solid whose base is the circle x^2 + y^2 = 4, where the cross sections perpendicular to the x-axis are all equilateral triangles.Β NOTE: Sometimes you'll be given isosceles right triangles or semicircles. Make sure you using the right formula!
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We find the side of the triangle by doing top minus bottom. Thus, the side of the triangle is 2√4 - x^2. Now, we use the formula for the equilateral triangle to find the area of the cross section (A = (side)2 *  √3 / 4). We can now find the volume by integrating the area function from x = -2 to x = 2. 
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