Find volumes of prisms, cylinders, cones, spheres, and pyramids.
Free, timed SHSAT practice sets on volume of 3d solids— every question includes a full worked explanation, and there's no account required.
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Volume measures the space inside a solid, in cubic units. Most solids follow one idea: area of the base \(\times\) height.
For an odd shape, break it into boxes: find the big volume, then subtract any cut-out part.
Common traps: forgetting the \(\dfrac{1}{3}\) for cones and pyramids (3 times too big), using the diameter where the formula needs the radius, and stopping at \(l \times w\), which is only the base area.
Volume \(= \pi r^2 h = \pi(13)^2(15) = 2535\pi\) cubic in. (Using the diameter, 26, in place of the radius gives \(10140\pi\). Forgetting to square the radius gives \(195\pi\). Doubling the radius instead of squaring it gives \(390\pi\).)
Volume \(= \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi(12)^2(15) = 720\pi\) cubic ft. (Leaving out the \(\dfrac{1}{3}\) gives the cylinder's volume, \(2160\pi\). Using the diameter, 24, in place of the radius gives \(2880\pi\). Forgetting to square the radius gives \(60\pi\).)
The full block has volume \(9 \times 6 \times 4 = 216\) cubic cm. The removed notch has volume \(4 \times 3 \times 4 = 48\) cubic cm. So the remaining solid is \(216 - 48 = 168\) cubic cm.