A pyramid is a 3D shape with a base, and triangular faces that meet at a single point at the top. Originally, they were the shapes of ancient pyramids in Egypt, and later inspired modern glass buildings.
It fact, the volume of a pyramid is exactly \(\frac{1}{3}\) the volume of a prism with the same base area and height. Intuitively, the prism can be thought of as three pyramids put together, so the pyramid has \(\frac{1}{3}\) the volume of the prism.
\begin{equation*}
\boxed{V_{\text{pyramid}} = \frac{1}{3} A h \quad \begin{cases} A \rightarrow \text{base area} \\ h \rightarrow \text{height} \end{cases}}
\end{equation*}
Example3.4.1.Volume of a Square Pyramid.
Find the volume of a square pyramid with base side 6 m and height 9 m.
For a regular square pyramid, the slant height is related to the height by the Pythagorean theorem. The distance from the center of the base to the midpoint of a base edge is \(\frac{a}{2}\text{.}\)
\begin{equation*}
SA = \begin{pmatrix} \text{area of all} \\ \text{triangular faces} \end{pmatrix} + \brac{\text{area of base}}
\end{equation*}
With a regular pyramid, the triangular faces are congruent, so it is enough to consider one of the faces. Each triangular face has base \(l\) (the side length of the base) and height \(s\) (the slant height). So the area of one face is \(\frac{1}{2} ls\text{.}\)
For a regular square pyramid, there are 4 faces, so the total area of the triangular faces is \(4 \cdot \frac{1}{2} ls\text{.}\) This area (not including the base) is called the lateral area, \(A_L\text{.}\) In general, if the base is a regular \(n\)-gon, then the lateral area is,
If the edge length is \(a\text{,}\) then each face has area \(\frac{\sqrt{3}}{4}a^2\) (this is a known formula for the area of an equilateral triangle), so,
\(V_{\text{prism}} = 10^2 \times 15 = 1500 \text{ cm}^3\text{.}\)\(V_{\text{pyramid}} = \frac{1}{3}(100)(15) = 500 \text{ cm}^3\text{.}\) The pyramid has exactly \(\frac{1}{3}\) the volume of the prism.
A right rectangular pyramid has base dimensions \(9.0 \text{ cm} \times w\) and height 10.0 cm. Its volume is \(68.4 \text{ cm}^3\text{.}\) Find \(w\text{.}\)
A wooden right rectangular pyramid has base dimensions 10.4 cm by 8.6 cm and height 14.8 cm. Find its volume to the nearest tenth of a cubic centimetre.
A desk puzzle is in the shape of a right rectangular pyramid with base 6 cm by 8 cm and height 10 cm. Calculate the amount of wood needed to create the puzzle.
A doghouse has a rectangular base (1.2 m \(\times\) 0.8 m) with a pyramidal roof. The roof has a height of 0.5 m above the top of the walls. Find the volume of the roof.
A model pyramid is constructed for a museum display. The total area of the four triangular faces is 3000 square inches, and the base has side length 50 in. Determine the height of the model to the nearest tenth of an inch.
Two glass pyramids are being compared. Pyramid A has a square base with side 35.0 m and height 20.6 m. Pyramid B has a square base with side 25.7 m and height 24.0 m. Which pyramid requires more glass for its four triangular faces?
A tent has the shape of a right square pyramid. It uses 4 poles, each 2.1 m long, as the lateral edges from the base corners to the apex. The base has side length 1.5 m.
The distance from the center of the base to a corner is \(\frac{1.5\sqrt{2}}{2} \approx 1.06\) m. The pyramid height is \(h = \sqrt{2.1^2 - 1.06^2} = \sqrt{4.41 - 1.12} = \sqrt{3.29} \approx 1.81\) m. The slant height is \(s = \sqrt{1.81^2 + 0.75^2} = \sqrt{3.28 + 0.56} = \sqrt{3.84} \approx 2.0\) m.
A pyramid-shaped building has a square base with side length 60 ft and height 38 ft. The four triangular exterior walls are to be coated with polished limestone. What area of limestone is needed, to the nearest square foot?
A glass pyramid greenhouse has a square base that measures 26 m on each side and a slant height of 35.4 m. How much glass is needed for the four triangular faces, to the nearest square metre?
A stone monument has the shape of a right square pyramid with slant height 1.6 m and base side length 0.8 m. Find the volume of the monument to the nearest tenth of a cubic metre.
The roof of a house is shaped like a right square pyramid with base 32 ft on each side. The attic must enclose a volume of at least \(4096 \text{ ft}^3\) of air. What is the minimum height of the roof?
Each face is a triangle with base 3.5 m and equal sides 4.5 m. The slant height is \(s = \sqrt{4.5^2 - 1.75^2} = \sqrt{20.25 - 3.06} = \sqrt{17.19} \approx 4.15\) m. Then \(h = \sqrt{4.15^2 - 1.75^2} = \sqrt{17.22 - 3.06} = \sqrt{14.16} \approx 3.8\) m.
A second ornament has the same shape and the same height, but a volume of \(96 \text{ in}^3\text{.}\) Find the side length of its base to the nearest inch.
A right rectangular pyramid has base dimensions 5 m by 3 m and a height of 10 m. A horizontal cut is made 2 m from the apex, and the smaller pyramid at the top is removed. Find the volume of the remaining piece (the frustum).
Full pyramid: \(V = \frac{1}{3}(15)(10) = 50 \text{ m}^3\text{.}\) The small pyramid has height 2 m. By similar triangles, its base is scaled by \(\frac{2}{10} = \frac{1}{5}\text{,}\) so the base is \(1 \times 0.6\) m. \(V_{\text{small}} = \frac{1}{3}(0.6)(2) = 0.4 \text{ m}^3\text{.}\) Remaining: \(50 - 0.4 = 49.6 \text{ m}^3\text{.}\)
A right pyramid has a base that is a regular hexagon with side length 5.5 cm. Each triangular face has two equal sides of length 7.5 cm. Find the surface area to the nearest square centimetre.
Subsection3.4.7Advanced: Derivation of Volume Formula
Consider a cube with side length \(x\text{,}\) and so of volume \(x^3\text{.}\) Consider the center point in the cube, and connect it to all 8 corners,
This divides the cube into 6 identical pyramids of equal volume --- one pyramid for each face of the cube. Each pyramid has its base on a face of the cube and its apex at the center. Here is one of the 6 pyramids highlighted (the one whose base is the bottom face),
Each pyramid has a base that is a face of the cube (area \(x^2\)), and a height equal to half the side length (\(\frac{x}{2}\)), since the center is in the middle of the cube.
