Derive the Constant, Do Not Recall It

You'll learnto build each cross-sectional area from the shape's own geometry, to tell a leg from a hypotenuse and a diameter from a radius in the wording, and to check a set of volumes against the heights of the shapes that produced them.

The method has not changed since Topic 8.7. What changes is the constant that turns a base length into an area, and every one of them is worth deriving rather than recalling. A semicircle built on a diameter s has radius s over 2, and the factor of 4 that follows is where most of this topic's lost points live. Two of the three wrong constants in circulation give answers that look entirely plausible, because they are off by a factor of two.

8 STEPS · 6 QUICK CHECKS · WHAT IS THE BASE TO THE SHAPE? · v1

equilateral s²√3/4 ⇒ right on a leg s²/2, on a hypotenuse s²/4 ⇒ semicircle πs²/8

Before you start

What you're looking at

The same two panels as Topic 8.7. On the left, the base region with one slice's footprint. On the right, that slice drawn face on — now as a triangle or a semicircle rather than a square.

The question

What is the base segment to this shape? Its side, one of its legs, its hypotenuse, its diameter, or its radius — each answer gives a different constant.

Watch for

Every figure here is drawn from its own geometry rather than scaled to a remembered area, so a shape that looks wrong beside the others is a constant that is wrong.

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