Volume of Pyramids

A **pyramid** is a 3D shape with one flat **base** (a polygon) and triangular faces that all meet at one point at the top, called the **apex**. The volume of every pyramid follows one rule: $$V = \frac{1}{3}Bh$$ Where: - $B$ = the **area of the base** (always find this first) - $h$ = the **perpendicular height**: the straight vertical distance from the apex down to the base, at a right angle to it ## Finding B for Different Bases | Base | Base area $B$ | Volume | |---|---|---| | Square, side $s$ | $B = s^2$ | $V = \frac{1}{3}s^2h$ | | Rectangle, $l$ by $w$ | $B = l \times w$ | $V = \frac{1}{3}lwh$ | | Triangle, base $b$, height $h_b$ | $B = \frac{1}{2} \times b \times h_b$ | $V = \frac{1}{3} \times \frac{1}{2}bh_b \times h$ | ## Why One Third? Take a prism and a pyramid with the **same base and the same height**. The pyramid holds exactly **one third** of the prism: you can pour three full pyramids of sand into the prism. So $$V_{\text{pyramid}} = \frac{1}{3}V_{\text{prism}} = \frac{1}{3}Bh$$ ## Height vs. Slant Height The **slant height** $l$ runs along a triangular face, from the apex to the middle of a base edge. It is longer than $h$. In a square pyramid, $h$, half the side and $l$ form a right triangle, so $$h^2 + \left(\frac{s}{2}\right)^2 = l^2$$

Square Pyramid

A pyramid has a square base with sides of $6$ cm and a perpendicular height of $10$ cm. What is its volume?

Find the base area: $B = s^2 = 6^2 = 36$ = $B = 36$ cm²

Multiply by the height: $B \times h = 36 \times 10 = 360$ = $360$ (this would be the prism)

Take one third: $V = \frac{1}{3} \times 360 = 120$ = $V = 120$ cm³

Answer: The volume is $120$ cm³.

Rectangular Base

A pyramid has a rectangular base of $8$ m by $6$ m and a height of $10$ m. Find its volume.

Find the base area: $B = l \times w = 8 \times 6 = 48$ = $B = 48$ m²

Multiply by the height: $48 \times 10 = 480$ = $480$

Divide by 3: $V = 480 \div 3 = 160$ = $V = 160$ m³

Answer: The volume is $160$ m³.

Triangular Base

The base of a pyramid is a triangle with a base of $6$ cm and a height of $4$ cm. The pyramid is $9$ cm tall. What is its volume?

Find the area of the base triangle: $B = \frac{1}{2} \times 6 \times 4 = 12$ = $B = 12$ cm²

Multiply by the pyramid's height: $12 \times 9 = 108$ = $108$

Take one third: $V = 108 \div 3 = 36$ = $V = 36$ cm³

Answer: The volume is $36$ cm³. Notice the two different heights: $4$ cm belongs to the base triangle, $9$ cm is the height of the pyramid.

Finding the Height From the Volume

A pyramid with a square base of side $9$ m has a volume of $324$ m³. How tall is it?

Find the base area: $B = 9^2 = 81$ = $B = 81$ m²

Multiply both sides of $V = \frac{1}{3}Bh$ by 3: $3V = Bh$, so $3 \times 324 = 972$ = $Bh = 972$

Divide by the base area: $h = \frac{3V}{B} = 972 \div 81 = 12$ = $h = 12$ m

Answer: The pyramid is $12$ m tall.

Using the Slant Height

A square pyramid has base sides of $10$ cm and a slant height of $13$ cm. Find its volume.

Draw the right triangle inside the pyramid: Legs: $h$ and half the side, $10 \div 2 = 5$. Hypotenuse: the slant height $13$ = $h^2 + 5^2 = 13^2$

Use Pythagoras to find h: $h^2 = 169 - 25 = 144$, so $h = \sqrt{144}$ = $h = 12$ cm

Find the base area: $B = 10^2 = 100$ = $B = 100$ cm²

Apply the volume formula: $V = \frac{1}{3} \times 100 \times 12 = \frac{1200}{3} = 400$ = $V = 400$ cm³

Answer: The volume is $400$ cm³. Using $13$ instead of $12$ would give a wrong answer of about $433$ cm³.

Pyramid shapes are everywhere, and their volume tells you how much space, air or material they hold: - **Ancient monuments**: the Great Pyramid of Giza holds about 2.6 million cubic meters of stone - **Modern buildings**: the glass pyramid of the Louvre in Paris has a volume of about $9000$ m³ of air to heat or cool - **Camping**: a pyramid tent's volume tells you how much space you have inside - **Roofs**: builders estimate attic space and the air to heat under a pyramid roof - **Packaging and food**: pyramid tea bags, chocolates and candles The same one-third idea also gives the volume of a cone, so learning it here helps you with every pointed solid.

Mistake: Forgetting the $\frac{1}{3}$ and calculating $V = Bh$

Why: $Bh$ is the volume of the prism with the same base and height. A pyramid fills only one third of it.

Correct: Always finish with $\div 3$: a square pyramid with $s = 6$ and $h = 10$ has $V = 360 \div 3 = 120$, not $360$.

Mistake: Using the slant height instead of the perpendicular height

Why: The slant height runs along a sloping face, so it is longer than the true height. Using it makes the volume too big.

Correct: Use the height that meets the base at a right angle. If only the slant height is given, find $h$ with $h^2 + \left(\frac{s}{2}\right)^2 = l^2$.

Mistake: Putting a side length in place of the base area

Why: $B$ is an area, not a length. Writing $\frac{1}{3} \times 6 \times 10$ for a square base of side $6$ leaves out a whole factor.

Correct: Compute $B$ first: $B = 6^2 = 36$, then $V = \frac{1}{3} \times 36 \times 10 = 120$.

Mistake: Forgetting the $\frac{1}{2}$ in a triangular base

Why: The area of a triangle is half of base times height, so $b \times h_b$ alone is twice too big.

Correct: For a triangular base with $b = 6$ and $h_b = 4$: $B = \frac{1}{2} \times 6 \times 4 = 12$, not $24$.

The Great Pyramid of Giza

Historians estimate how much stone the Egyptians moved by treating the Great Pyramid as a square pyramid.

When it was built, the pyramid had a square base of about $230$ m per side and was about $147$ m tall.

Pyramid Tents

Camping gear makers give the inside volume of tents so you can compare how roomy they are.

A tent with a square floor of $2$ m by $2$ m and a center pole of $1.5$ m is a square pyramid.

Pyramid Roofs

Heating engineers need the volume of the air space under a roof to size a heater or a ventilation fan.

A pyramid roof (hip roof that ends in one point) sits on a house with a rectangular floor plan.

The volume of any pyramid is $V = \frac{1}{3}Bh$, where $B$ is the base area and $h$ the perpendicular height

Find $B$ first: $s^2$ for a square, $l \times w$ for a rectangle, $\frac{1}{2} \times b \times h_b$ for a triangle

A pyramid holds exactly one third of the prism with the same base and height

Use the perpendicular height, not the slant height; if needed, find it with $h^2 + \left(\frac{s}{2}\right)^2 = l^2$

To find the height from the volume, use $h = \frac{3V}{B}$

Volume is always in cubic units (cm³, m³)

Q: Why is there a $\frac{1}{3}$ in the formula?

A: A pyramid narrows to a point, so it holds less than a prism with the same base and height. Experiments with sand or water show that exactly three pyramids fill the prism. It can also be proven: a cube can be cut into three identical pyramids that each have one face of the cube as their base.

Q: Is the cone formula related?

A: Yes. A cone is like a pyramid with a circular base, so $V = \frac{1}{3}Bh$ with $B = \pi r^2$ gives $V = \frac{1}{3}\pi r^2 h$, the same one-third rule.

Q: Does the formula work if the apex is not above the center of the base?

A: Yes. A leaning (oblique) pyramid has the same volume as a straight one with the same base and the same perpendicular height. Only measure $h$ straight up from the level of the base to the apex.

Q: What if the base has 5 or 6 sides?

A: The formula $V = \frac{1}{3}Bh$ works for any base. The only change is how you find $B$: split the polygon into triangles or use its area formula, then multiply by $h$ and divide by 3.

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Volume of Pyramids

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Measurement / Volume & Capacity

Volume of Pyramids

Learn how to find the volume of square, rectangular and triangular pyramids with V = (1/3)Bh, and how to get the height from the slant height.

Advanced25 min

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