Problem
Angles between segments drawn on a cube
A cube has its top and bottom corners labelled. Nothing is drawn on the cube itself. The segments have to be imagined. Find the size of each of the three named angles.
Reasoning · Grade 5-2 Exploring the Cube
Your answer
Watch out Slanted angles invite guessing, but equal face diagonals pin them down exactly.
How to solve
Strategy Draw a Diagram — Each question names three vertices, and three points always make a triangle, so the plan is the same three times: draw the triangle onto the cube, work out what kind of triangle it is, and read the angle off. To classify the triangle I only need to sort its three sides into edges, face diagonals and cube diagonals, because all the edges are equal to one another and all the face diagonals are equal to one another. That turns a three-dimensional question into an ordinary flat-triangle question, which is the whole point of splitting it into subproblems. Building or holding a cube and stretching a piece of string between the named corners makes each triangle real, and it keeps the distorted picture from misleading me.
1STEP 1
Sort out the three possible lengths first
A cube has only edge, face diagonal and body diagonal.
edge ＜ face diagonal ＜ cube diagonal
This one observation replaces all measuring. Since two sides labelled the same way must be equal, you can spot an isosceles or an equilateral triangle without knowing a single number.
K.G.B.4Identify Subproblems2STEP 2
(1) Locate triangle FHG — it lies flat in the bottom face
The first triangle lies flat in the bottom face.
GF = GH = edge, ∠ FGH = 90°
Three corners of one face never leave that face, so the hard part of the problem — imagining a slanted triangle in space — simply does not arise here. It is an ordinary square-corner question from Grade 3.
3.G.A.1Draw A Diagram3STEP 3
(1) Finish: the base angles of a right isosceles triangle are 45°
Right isosceles, so the base angle is 45 degrees.
∠ GFH = ∠ FHG = (180° - 90°)/2 = 45°
Cutting a square along its diagonal is something you can do with a folded piece of paper: the two halves match, so the two sharp corners are the same, and half of the leftover 90° is 45°.
8.G.A.5Identify Subproblems4STEP 4
(2) Notice that GH sticks straight out of the face that contains BG
In the second an edge sticks straight out of a face.
GH ⊥ face BCGF, GB ⊂ face BCGF
Stand a pencil upright on a table: it makes a right angle with every line you draw on the table, not just with the two edges of the table. Edge GH is that pencil and the front face is that table.
4.G.A.2Visualize Spatial Relationships5STEP 5
(2) Read off the right angle
So that angle is 90 degrees.
∠ BGH = 90°
The picture is drawn at a slant, so on the page this angle looks nothing like 90°; that is exactly why the reasoning has to come from the cube's right-angled faces rather than from the drawing.
4.G.A.2Draw A Diagram6STEP 6
(3) Check the three sides of triangle AFH
In the third all three sides are face diagonals.
AF, FH, AH are diagonals of faces ABFE, EFGH, ADHE
Triangle AFH is exactly the cut you would make to slice the corner E off the cube, and every such corner slice cuts three faces along one diagonal each.
K.G.B.4Visualize Spatial Relationships7STEP 7
(3) Finish: the triangle is equilateral, so every angle is 60°
Equilateral, so the angle is 60 degrees.
AF = FH = AH → ∠ AFH = 180°/3 = 60°
This is the surprise of the whole page: a triangle drawn across three different faces of a cube, which looks lopsided in the picture, is in fact perfectly equilateral — because the only lengths involved are three copies of the same face diagonal.
8.G.A.5Identify SubproblemsTriangle AFH has three equal sides, so every one of its angles is 60 degrees.
Why?
Equal sides face equal angles, so three equal sides force all three angles to be the same size.
🧱Equal sides face equal anglesFold an isosceles triangle down its middle: the two equal sides land on each other and carry the two base angles together, so those angles are equal — and the reverse holds too.
Why?
The three angles add to one straight angle, so three equal shares of 180 degrees is 60 degrees each.
🧱A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.
8STEP 8
Check all three with a real cube
A real cube confirms all three.
Angles in a slanted picture are the easiest thing in geometry to misjudge, so touching the real solid is the check that matters most here.
4.MD.C.5Create A Physical RepresentationAnswer
45, 90, 60 degrees
180 − 90 = 90, 90 ÷ 2 = 45
All three answers are between 0° and 180°, as any angle of a triangle must be, and none of them depends on the size of the cube, which is right because no length was given. Each also sits sensibly inside its own triangle: triangle FHG is right-angled at G, so 45° + 45° + 90° = 180°; triangle BGH has the cube diagonal BH as its longest side, and the largest angle of a triangle must be opposite the longest side, so the 90° belongs at G and not at B or H; triangle AFH has three equal sides and 60° + 60° + 60° = 180°. The three answers are also the only three sizes this kind of question can produce — 45°, 60° and 90° — because a triangle on the corners of a cube can only be right isosceles, equilateral, or the edge-plus-face-diagonal right triangle of part (2).
Takeaway
On a cube there are only three lengths - edge, face diagonal, cube diagonal - so just naming the sides of your triangle tells you whether the angle is 45°, 60° or 90°.
- Sort out the three possible lengths first
- (1) Locate triangle FHG — it lies flat in the bottom face
- (1) Finish: the base angles of a right isosceles triangle are 45°
- (2) Notice that GH sticks straight out of the face that contains BG
- (2) Read off the right angle
- (3) Check the three sides of triangle AFH
- (3) Finish: the triangle is equilateral, so every angle is 60°
- Check all three with a real cube