At a glance. One lesson of about 60 minutes. It assumes Pythagoras' theorem (Unit 2) and the area of a triangle. Students find the volume of a pyramid, the slant height and the surface area of a square pyramid, and use Pythagoras both ways round. Every length is a whole number, so no calculator is needed.
Calculator: no calculator. No calculator is needed: every triangle is a whole-number triple (3, 4, 5; 5, 12, 13; 7, 24, 25; 8, 15, 17; 9, 12, 15; 12, 16, 20) and the thirds divide exactly. The thinking is the structure.
What students will learn
Know
- The volume of a pyramid is ⅓ × base area × perpendicular height.
- The slant height of a square pyramid satisfies s² = h² + (b/2)².
- The whole surface is the base plus four triangles of base b and height s.
Understand
- Three pyramids fill the prism with the same base and height, so one is a third; the point does not make it a half.
- The height and the slant height are different lengths for different jobs: volume and face area.
- Pythagoras can be used to find a hypotenuse or a leg, whichever is missing.
Do
- Find the volume of a pyramid from its base and height.
- Find the slant height and the surface area of a square pyramid.
- Work back to a height from the volume, or to the height from the slant height.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the volume and surface area formulas and Pythagoras (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Ask students to predict the volume of the pyramid for a new base and height before they move the sliders, and to explain why it is always one third.
- Criterion C. Communicating: drawing the right-angled triangle with its three lengths labelled before calculating, as in the worked example.
- Criterion D. Applying mathematics in real-life contexts: a museum, a glass pyramid and a market stall roof (the hook, the worked example and question 7).
The one thing to do with the figure
Settle the hook with the first figure, then use the second for the triangle. Ask for the number of pyramids that fill the box before moving the sliders, then move them: it is always three, whatever the base and height.
In the second figure, stop on the base 24 and height 5 pyramid. The slant height is 13 but the height is only 5. Ask which one goes into the volume and which into the faces.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook: the box of sand and the pyramid. Collect predictions of how many pyramids fill the box. |
| 10 min | First figure: move both sliders, see the third. Settle the hook. |
| 10 min | Key idea; second figure and the right-angled triangle; the glass pyramid worked example; students finish the 8 and 15 pyramid in pairs. |
| 5 min | Question 1, the check. If many choose A or C, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 120 cm³; 10 cm; 21. |
|---|---|
| Hook | Three pyramids of sand fill the box. |
| Worked example | s = 13 m; V = 400 m³; faces 260 m². |
| Faded example | 320 cm³. |
| 1. Which expression is the volume? | B. A uses the side instead of the base area; C is the prism; D uses a half. |
| 2. Volume of a square pyramid | 24. ⅓ × 9 × 8. |
| 3. A rectangular base | 100. ⅓ × 30 × 10. |
| 4. Slant height from the height | 25. 24² + 7² = 625. |
| 5. Whole surface from the slant height | 920. 400 + 4 × 130. |
| 6. Working back to the height | 18. 3 × 150 ÷ 25. |
| 7. Canvas for a tent roof | 540. s = 15; 4 × 135. |
| 8. Volume from the slant height | 2400. h = 8; ⅓ × 900 × 8. |
| 9. The perpendicular height | C. A subtracts lengths; B uses the whole base; D adds. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the side not the area, the prism, or a half. | Ask: what is the base area? Then, what is the volume of the prism around it? |
| 72, 8 or 36 in question 2 | Gave the prism, the side instead of the area, or used a half. | Hold up a box and a pyramid with the same base: how many pyramids fill the box? |
| 300, 150 or 10 in question 3 | Gave the prism, used a half, or left out the height. | Write the three numbers to multiply on the board, then divide by 3. |
| 31, 17 or 625 in question 4 | Added, subtracted, or forgot the square root. | Draw the triangle: which side is the hypotenuse? What does the theorem say? |
| 520, 1440 or 530 in question 5 | Left out the base, missed the half, or counted one face. | List the faces: one square and how many triangles? |
| 6, 2 or 450 in question 6 | Forgot to undo the third, divided by 3 as well, or did not divide by the base. | Say it in words: a third of the base times the height is 150. Undo the steps in reverse. |
| 432, 135 or 864 in question 7 | Used the height not the slant height, one face, or added the base. | Draw the right-angled triangle with half the base and the height, find the slant height, then ask which length lies along the canvas. |
| 7200, 15300 or 5100 in question 8 | Gave the prism, or used the slant height as the height. | Ask which length is straight up inside the pyramid. Draw the right-angled triangle. |
| A, B or D in question 9 | Subtracted, used the whole base, or added. | Mark half the base on the triangle and ask what the hypotenuse is. |
Questions worth asking
- Why is a pyramid one third and not one half? A cube splits into three identical pyramids that share one corner as their apex. Each has a face of the cube as its base and height b, so each is ⅓b³; stretching the cube stretches all three together, so the third stays. Listen for “three identical pyramids fill the cube”, with the first figure as the evidence for other shapes.
- Which is longer, the height or the slant height? The slant height: it is the hypotenuse of the triangle that has the height as a leg.
- How would the surface area change for a pyramid with a rectangular base? The two pairs of faces have different slant heights, so the work is longer, though the idea is the same.
- What is the volume of a pyramid with the same base and height but its apex over a corner? The same: the one-third rule depends on the base and the height, not on where the apex is.
Differentiation
A way in for anyone stuck
- Provide a card with the right-angled triangle drawn inside the pyramid and the three lengths named.
- Give students the Pythagorean triples to look for: 3, 4, 5; 5, 12, 13; 7, 24, 25; 8, 15, 17; 9, 12, 15.
- Use questions 1 and 2 with a model: fill a pyramid with sand or rice and pour it into a box.
Stretch
- Question 9, then: find the volume of the pyramid. (⅓ × 400 × 24 = 3200 cm³.)
- A square pyramid has base side b and height h. Write a formula for its slant height, then for its surface area, and test it on 6 and 4.
Students learning in English as an additional language
“Slant” means leaning, not straight up. “Apex” is the top point. “Perpendicular height” is the straight-up height, at right angles to the base. Have students say “height” for the straight line inside and “slant height” for the line on the face.
| Word | Plain meaning | Example |
|---|---|---|
| pyramid | a solid with a flat base and sloping faces that meet at a point | a square pyramid |
| apex | the top point where the sloping faces meet | the tip of a pyramid |
| perpendicular height | the height straight up from the base to the apex | 12 m |
| slant height | the height of a sloping face, measured up the middle of the face | 13 m |
| face | one flat surface of a solid | a triangle on a pyramid |
| Pythagoras' theorem | in a right-angled triangle, the squares of the two short sides add to the square of the longest | 12² + 5² = 13² |
Sentence starters for explaining aloud or in writing:
- The pyramid holds a third of the prism because …
- The right-angled triangle has sides …, … and …
- I use the height for the … and the slant height for the …
Two things not to say
“Use the formula, it is a third.” Show the three pyramids fill the prism first, and then name the rule.
“Height and slant height are about the same.” They are different lengths and mixing them is the main error.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- VolumeA square pyramid has base side 6 cm and height 5 cm. Find its volume.
60 cm³. - SlantHalf the base is 12 cm and the height is 5 cm. Find the slant height.
13 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA pyramid-shaped tent has a square base of side 6 m and a height of 4 m. Find the slant height and the canvas for the four sloping faces.
Half the base is 3, so s = 5 m, and the canvas is 4 × ½ × 6 × 5 = 60 m². - Earlier work (Year 8)A triangular prism has a right-angled triangle end with legs 3 cm and 4 cm and is 10 cm long. Find its volume.
½ × 3 × 4 × 10 = 60 cm³.
3Challenge
Reasoning, generalizing and working backwards.
- Working backA square pyramid has volume 96 cm³ and height 8 cm. Find the base side.
Base area 36, so the side is 6 cm. - ReasoningA square pyramid's base side is doubled and its height stays the same. What happens to the volume?
It is multiplied by 4.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? A hollow pyramid and a hollow box with the same base and height, and a cup of rice, show the one-third rule in a minute.