At a glance. One lesson of about 60 minutes. It assumes Pythagoras and sine, cosine and tangent. Students find floor and space diagonals, sloping edges of pyramids, and angles between a line and the base, and work back from a diagonal to an edge.
Calculator: GDC optional. Square roots of sums and inverse tangents need a calculator, so the label is GDC optional. The optional section is about entering a root of a sum in one go.
What students will learn
Know
- The space diagonal of a cuboid is √(l² + w² + h²).
- The angle between a line and a plane is the angle with its shadow on the plane.
- A sloping edge of a square-based pyramid has half the base diagonal as the shadow.
Understand
- Every 3D length or angle in this unit sits in some flat right-angled triangle.
- A shape's flat triangle may need another triangle first, as with the floor diagonal.
Do
- Draw and label the flat triangle inside a solid.
- Use Pythagoras twice for a space diagonal.
- Find an angle between a line and a plane.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: Pythagoras and trigonometry applied to solids (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students find which edge sets give whole-number space diagonals and how the edge angle changes with height.
- Criterion C. Communicating: separate, labelled flat sketches for each triangle used (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the ruler in the box, the room (questions 2 and 3) and the storeroom pole (question 9).
The one thing to do with the figure
Let students see the two triangles. In the first figure, ask students to name the legs and hypotenuse of the floor triangle, then of the second triangle, before they read the text.
In the second figure, ask: if you double both b and h, what happens to the angle? (It stays the same: a similar pyramid.)
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: does the ruler fit? Collect votes. |
| 12 min | First figure; key idea; settle the hook with worked example 1; the warning card. |
| 10 min | Second figure; worked examples 2 and 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the idea of a hypotenuse. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 13 cm; 10 cm; 11°. |
|---|---|
| Hook | No: the longest line is about 27.7 cm, shorter than 30 cm. |
| Worked examples | 27.7 cm; 19.0°; 11.7 cm and 43.3°. |
| Faded example | 65 cm. |
| 1. Space diagonal of a cuboid | D. A adds the edges; B adds a leg to a hypotenuse; C omits one edge. |
| 2. Floor diagonal | 15 m. |
| 3. Space diagonal from a floor diagonal | 17 m. |
| 4. Space diagonal, decimal answer | 12.9 cm. √166. |
| 5. Angle between a diagonal and the base | 26.6°. tan−1(5 ÷ 10). |
| 6. Sloping edge of a pyramid | 15.6 cm. √242. |
| 7. Angle of a pyramid edge | 27.9°. tan−1(6 ÷ 11.31). |
| 8. Working back to a height | 7 cm. |
| 9. Longest pole in a room | B. A adds the edges; C and D each leave one edge out. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Added edges, added a leg to a hypotenuse, or left out an edge. | Ask for two sketches: the floor triangle, then the vertical triangle. |
| 21 or 3 in question 2 | Added or subtracted the sides. | Write a² + b² = c² next to the sketch first. |
| 23 or 15 in question 3 | Added the legs, or forgot the height. | Ask: which two lengths are at right angles to each other now? |
| 22 or 18.2 in question 4 | Added the edges, or added the height after the first root. | Write all three squares under one root. |
| 32.0 or 63.4 in question 5 | Used one edge as the shadow, or inverted the ratio. | Name the line, its shadow and the vertical on the sketch. |
| 13.9 or 23.2 in question 6 | Used half the base edge, or the whole diagonal. | Ask: where is the foot of the height, and how far is it from a corner? |
| 36.9 or 20.6 in question 7 | Used half the edge, or the whole edge, as the shadow. | Draw the cross-section through the diagonal, with its half marked. |
| 9.2 or 49 in question 8 | Took off one edge only, or forgot the final root. | Write 11² = 6² + 6² + h² before rearranging. |
| A, C or D in question 9 | Added, or left an edge out. | Count three squares on the page before the root. |
Questions worth asking
- Why is the space diagonal always shorter than the sum of the three edges? Listen for “the straight line is the shortest path” or “a hypotenuse is shorter than the two legs added”.
- Where does the foot of a pyramid's height sit on the base? At the centre, where the diagonals cross.
- Which triangle did you draw first, and why? The one whose sides you could already find.
- Why does a smaller copy of a pyramid keep its angles? Similar solids have equal angles: all lengths scale by one factor.
Differentiation
A way in for anyone stuck
- Give a card cuboid or a box to hold: students trace the floor diagonal with a straw, then the space diagonal.
- Provide a frame: “floor diagonal² = ____ + ____; space diagonal² = ____ + ____ = ____”.
- Restrict to cuboids with whole-number space diagonals (2, 3, 6 and 2, 6, 9 and 4, 4, 7) at first.
Stretch
- Show that the space diagonal of a cube of side a is a√3, and find the angle between it and the base.
- A square-based pyramid has all eight edges 10 cm long. Show that its vertical height is 5√2 cm.
Students learning in English as an additional language
“Diagonal” is a line joining two corners that are not neighbours. “Space diagonal” goes through the inside of a solid. “Shadow” (projection) is the line on the floor straight below a line in the air. Hold a pencil above a desk to show the shadow.
| Word | Plain meaning | Example |
|---|---|---|
| space diagonal | a line from one corner of a solid to the opposite corner, through the inside | the longest rod that fits in a box |
| cuboid | a solid with six rectangular faces | a shoebox |
| floor diagonal | the diagonal of the base of a solid | the line across the floor of a room |
| sloping edge | an edge of a pyramid from the apex to a corner of the base | the edge from the top of a pyramid-shaped tent to a corner peg |
| projection | the shadow of a line on a plane, straight below it | half the base diagonal, under a sloping edge of a pyramid |
| apex | the top point of a pyramid | the tip of a roof |
Sentence starters for explaining aloud or in writing:
- The flat triangle I use has legs … and … and hypotenuse …
- The shadow of the line on the base is …
- The angle is the inverse tangent of … divided by …
Two things not to say
“Just add them.” The edges are at right angles: square them and add.
“Use half the base.” That is for a face; an edge uses half the diagonal.
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.
- Space diagonalFind the space diagonal of a cube of side 5 cm, to 1 decimal place.
8.7 cm. - Earlier work (Year 9)A right-angled triangle has short sides 7 and 24. Find the hypotenuse.
25.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cable runs from a floor corner to the opposite ceiling corner of a hall 5 m by 4 m by 3 m. How long is it, to 1 decimal place?
7.1 m. - Working backA cuboid has a 4 cm by 7 cm base and a space diagonal of 9 cm. How high is it?
4 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the space diagonal of a cube of side a is a√3.
Floor diagonal a√2; then √(2a² + a²) = a√3. - OpenFind a cuboid with whole-number edges whose space diagonal is a whole number, other than 2, 3, 6 and 4, 4, 7.
For example 2, 10, 11 (diagonal 15) or 1, 12, 12 (diagonal 17).
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. “Show me the steps” gives the same states in words.
No device? A shoebox and a piece of string show the space diagonal. The With a GDC section is optional.