Measures and solids · Ages 12–13
3-D figures
From “Prisms, pyramids and round solids” to “Euler's formula and regular polyhedra”, step by step, each skill with its methods, worked examples and answers.
Skills and methods in this topic, easy to hard
- Prisms, pyramids and round solidsPrism: n + 2 faces, 3n edges, 2n vertices; pyramid: n + 1, 2n, n + 1
MethodA hexagonal prism's edges: 6 top + 6 bottom + 6 upright = 18
Another way: Count from the facesHexagonal prism: (6 × 4 + 2 × 6) ÷ 2 = 18
- Cross-sectionsPrisms: same as the base; pyramids: smaller near the top; sides = faces cut
MethodA cylinder cut through its axis: a rectangle, the height by the diameter
Another way: Count the faces cutThrough 3 of a cube's faces: a triangle
- Nets of solidsCount the faces; in a cube's net, squares one apart in a line are opposite
MethodA pentagonal prism's net: 5 rectangles + 2 pentagons
Another way: Fold it in your headKeep one face down and fold the others up one at a time
- Drawing solids on paperHidden edges dashed; slanting edges half length
MethodA cuboid: one hidden corner, 3 dashed edges
Another way: Isometric drawingUpright edges stay upright, lengths true
- Three viewsFront: each column's tallest; top: where there are cubes
MethodA cone: a triangle from the front, a circle with its centre dot from above
Another way: Back from the viewsEach place at most the smaller of its column's and row's tallest
- Euler's formula and regular polyhedraF + V − E = 2; edges = all the faces' sides ÷ 2
MethodA cube: 6 + 8 − 12 = 2
Another way: Count from the facesIcosahedron: 20 × 3 ÷ 2 = 30 edges
Worked examples
From different skills in this topic. Have a go first, then open each one for the answer and the working.
Example 1Prisms, pyramids and round solids →
How many edges does a prism on a 6-sided base have?
Show the answer and working
Answer: 18 edges
6 round the top, 6 round the bottom and 6 upright: 6 × 3 = 18.
Example 2Drawing solids on paper →
The solid is built of equal cubes, every column standing on the ground. How many cubes are there?
Show the answer and working
Answer: 11 cubes
Count each column's height, hidden cubes too: front 4 + 0 + 0, back 4 + 2 + 1, 11 in all.
Example 3Euler's formula and regular polyhedra →
A polyhedron has 8 faces and 12 edges. By Euler's formula, its vertices V = 2 − + = .
Fill in each blank
Show the answer and working
Answer: 8, 12, 6
F + V − E = 2 rearranges to V = 2 − F + E = 2 − 8 + 12 = 6.
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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