How do I find the volume of a pyramid?
Volume is ⅓ × base area × height. Keep every measurement in the same unit before multiplying — mixing centimetres and metres is the most common source of an answer that is out by a factor of a thousand or a million. Volume units are cubic, so converting afterwards means cubing the conversion factor: 1 m³ is 1,000,000 cm³, not 100.
How do I find the surface area of a pyramid?
Surface area is base area plus the slant faces. Work out each face separately and add them, rather than trying to apply a single remembered formula — it is slower but far more reliable, and it makes it obvious when a face should be excluded because it is not exposed.
Why is a pyramid's volume exactly one third of the prism?
Because three identical pyramids can be assembled to fill a prism with the same base and height — a result provable by dissection for some cases and by calculus in general. The same one-third relationship holds between a cone and its cylinder. It is one of the tidier results in solid geometry and worth demonstrating physically if you can.
What is the difference between height and slant height?
Height is the perpendicular distance from the apex straight down to the base. Slant height runs along the middle of a triangular face from apex to base edge, and is always longer. Volume uses the perpendicular height; surface area of the faces uses slant height. Substituting one for the other is the most common pyramid error, and it inflates the volume noticeably.
How do I find slant height from the perpendicular height?
By Pythagoras, using half the base width as the other leg: slant = √(h² + (b/2)²) for a square pyramid. For the Great Pyramid at 146 m tall with a 230 m base, the slant height is about 186 m — which is why the faces look far longer than the height suggests.
Why does my answer differ from the textbook's?
Usually one of three things. Rounding too early — carry full precision through the working and round only the final answer. Mixed units, which is the largest source of error. Or using a rounded value of π: use your calculator's π rather than 3.14, which introduces an error of about 0.05% that compounds when cubed.