At a glance. One lesson of about 60 minutes. It assumes Pythagoras and the three ratios. Students find a space diagonal, the angle of a line with a plane, and the angle between two planes in a cuboid and a pyramid, and prove that a face is steeper than an edge.
Calculator: GDC optional. A calculator is used for roots and inverse tangents, so the label is GDC optional. A scientific calculator does the same work, and the closed section covers keeping an exact root until the last step.
What students will learn
Know
- The longest diagonal of a cuboid is √(l² + w² + h²).
- The angle between a line and a plane is the angle between the line and its shadow on the plane.
- The angle between two planes is measured with a line in each plane perpendicular to the join.
Understand
- A 3D problem is a chain of right-angled triangles, each standing on a line in the previous one.
- The face's horizontal leg and the edge's horizontal leg are different, so the two angles differ.
Do
- Find a space diagonal and the angle it makes with a face or base.
- Find the angle between two planes by naming the two perpendiculars.
- Prove a general statement about angles in a pyramid.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using Pythagoras and tangent in the right triangle, and naming the correct angle (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure. Students predict how face and edge angles change with base and height and say why the face is always steeper.
- Criterion C. Communicating: a labelled sketch of the solid and a separate flat triangle for each step; the proof in worked example 3.
- Criterion D. Applying mathematics in real-life contexts: the steel container and the store room in Lagos.
The one thing to do with the figure
Make students predict, then reveal. Before the first figure, ask each student to sketch where the longest bar lies. Then raise and lower the height and ask which edge of the container's triangle changes.
In the second figure, set the base to 12 and the height to 12, then ask: which is steeper, the face or the edge? Move the sliders to try to make the edge steeper, and say why it cannot happen.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the 14 m bar. Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to which line is the shadow. |
| 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; 36.9°; 9.8 cm. |
|---|---|
| Worked examples | 11.2 m and 26.6°; 66.0° and 57.8°; the proof compares a/2 with a√2/2. |
| Faded example | 15.6°. |
| Something to think about | 56.3°, 26.6° and 25.4°: long-side faces, short-side faces, then edges. |
| 1. Setting up an angle in a cuboid | C. A uses a side of the base; B treats the base diagonal as the hypotenuse; D uses the space diagonal as adjacent. |
| 2. Longest rod in a room | 8.6 m. √74. |
| 3. Height from the angle of a diagonal | 8.2 cm. √136 × tan 35°. |
| 4. Angle of a pyramid's face with the base | 54.5°. tan−1(7 ÷ 5). |
| 5. Sloping edge of a rectangular pyramid | 11.5 cm. √133. |
| 6. Angle between a face and the base of a cuboid | 38.7°. tan−1(4 ÷ 5). |
| 7. Angle between a line and a plane that is not the base | 26.9°. tan−1(5 ÷ √97). |
| 8. Face angle from a given edge | 64.1°. h = √153, then tan−1(h ÷ 6). |
| 9. Naming the angle between two planes | D. A puts the angle at the wrong vertex, O; B is the other face's angle; C is inside the face. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Used a side of the base, treated the base diagonal as the hypotenuse, or used the space diagonal as adjacent. | Ask which line is the shadow of the diagonal on the base, and draw that triangle standing on it. |
| 8.1 or 14 in question 2 | Stopped at the floor diagonal, or added the edges. | Sketch the two triangles: one flat on the floor and one standing on its diagonal. |
| 7.0 or 6.7 in question 3 | Used the length as the horizontal leg, or took the base diagonal as the hypotenuse. | Mark the shadow of the diagonal on the base in colour and say which side of the triangle it is. |
| 35 or 44.7 in question 4 | Used the whole side, or the half diagonal. | Ask: which point of the base edge does the face's perpendicular touch? |
| 17 or 10.8 in question 5 | Used the whole base diagonal, or half the longer side. | Mark the centre of the base, then the corner, and find the distance between them. |
| 51.3 or 24 in question 6 | Swapped the legs, or used AB. | Mark the join in colour and ask which edges are perpendicular to it. |
| 29.1 or 21.2 in question 7 | Used AB as the shadow, or the angle with the base. | Ask: onto which plane is G dropped, and where does it land? |
| 55.6 or 66.4 in question 8 | Gave the edge's angle, or used the edge as a slant. | Name the triangle: apex, centre, midpoint of AB. |
| A, B or C in question 9 | Put the angle at the centre O, named the angle of the other face, or an angle inside the face. | Ask: along which line do the face VBC and the base meet? Draw one line in each plane perpendicular to it. |
Questions worth asking
- Why is the longest bar in a box longer than any edge? Listen for “it is the hypotenuse of a triangle that stands on the floor diagonal”.
- Where is the shadow of a line on a plane, and how do you find it? Listen for “drop a perpendicular from each end”.
- Why are two lines perpendicular to the join the right lines for the angle between two planes? Listen for “so the angle does not depend on where along the join you measure”.
- Does the proof in worked example 3 depend on the height? No: the height is the same for face and edge and cancels.
Differentiation
A way in for anyone stuck
- Give a net or a ready-drawn solid with the flat triangle already outlined for the first two questions.
- Provide a template: shadow, perpendicular, horizontal leg, vertical leg, ratio.
- Restrict to the first four questions at first, using whole-number results where possible.
Stretch
- A rectangular pyramid has a base a by b and height h. Write the angle of each of the four faces with the base and say which pair of faces is steeper when a > b.
- Show that the angle between a space diagonal and the base of a cube is tan−1(1 ÷ √2).
Students learning in English as an additional language
“Plane” means a flat surface that goes on forever; a face is a piece of a plane. “Shadow” here means the line you get by dropping each point straight down. Use a book on a table to show the angle between two planes.
| Word | Plain meaning | Example |
|---|---|---|
| space diagonal | a line from one corner of a solid to the opposite corner, through the inside | the longest rod in a box |
| plane | a flat surface that goes on in every direction | a table top, thought of as endless |
| perpendicular | at a right angle to | a wall meets the floor at a perpendicular |
| shadow of a line | the line you get by dropping each point of it straight onto a plane | a stick's shadow at noon |
| slant | the line up the middle of a sloping face, perpendicular to the base edge | the line a ball would roll down a roof |
Sentence starters for explaining aloud or in writing:
- The angle is in the triangle that stands on …
- The horizontal leg is … because the apex is above …
- The two planes meet along …, so I draw a line in each at right angles to it.
Two things not to say
“The angle is just the one at the corner.” Name the two lines and the plane first.
“Use the edge.” An edge is not the slant of a face.
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.
- DiagonalA cuboid is 3 cm by 4 cm by 12 cm. Find the longest diagonal.
13 cm. - Earlier work (Year 9)Find the height of an equilateral triangle of side 10 cm, to 1 decimal place.
8.7 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- PyramidA square-based pyramid has base 6 cm and height 4 cm. Find the angle of a face with the base, to the nearest degree.
53°. - EdgeThe same pyramid: find the length of a sloping edge, to 1 decimal place.
5.8 cm.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why the angle between a space diagonal and the base of a cuboid gets closer to 90° as the height grows, but never reaches it.
tan of the angle is height ÷ base diagonal, which grows without limit, but the base diagonal is never zero, so the angle stays below 90°. - OpenA cuboid has a square base. For which height is the angle between a space diagonal and the base exactly 45°?
When the height equals the base diagonal, a√2.
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 book on a desk and a pencil show shadows and the angle between two planes. The With a GDC section is optional.