Similar shapes – enlargement, scale factor k, k² and k³
MathematicsPlane GeometryAges 13–14
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Sign in to playFour screens on similarity. Enlargement: drag the centre O and the vertices, choose a scale factor k (including fractions and negative values) and see sides multiplied by |k|, angles unchanged and area multiplied by k². Area & volume: enlarge a cuboid built from unit cubes (2D or 3D view) and count squares and cubes to see k² and k³. Similar triangles: find missing sides, check your answers and view the worked solution. Map scales: measure distances and areas on a map and convert them to real life.
Lesson: Similar shapes; enlargement (dilation) with a centre; scale factor, area and volume factors; map scales
What it shows
An enlargement with centre O and scale factor k sends each point A to A′ with OA′ = k·OA along the line OA; for k < 0 the image lies on the other side of O, turned through 180°. All lengths are multiplied by |k| and angles are unchanged, so the image is similar to the original. Areas are multiplied by k² and volumes by k³, as the unit squares and cubes show. Similar triangles have equal angles, so one known pair of corresponding sides gives k. A map is a reduction with scale 1 : n, so areas scale by n². Points snap to whole grid units.
How to use
On Enlargement, move the k slider or press a preset such as 1/2 or −1, drag O and the vertices, and compare the table of sides, angles, perimeter and area. On Area & volume, change k and switch to 3D. On Similar triangles, type x and y, press Check, then Show solution or New problem. On Map scales, choose a Scale and drag P, Q and the forest corner.
Parameters you can change
- Screen Enlargement, Area & volume, Similar triangles, Map scales
- Scale factor k -3–3
- Original shape Triangle, Quadrilateral, L-shape
- Show rays from the centre O
- Scale factor k for the cuboid 0.5–3
- Solid Cube 2 × 2 × 2, Cuboid 4 × 2 × 2
- Triangle problem type Two separate triangles, DE parallel to BC
- Map scale 1 : 1 000, 1 : 10 000, 1 : 25 000, 1 : 50 000, 1 : 100 000
Questions to explore
- What happens to the image when k changes from 2 to −2? Which measurements stay the same?
- If a model is enlarged by k = 3, how many times larger are its surface area and its volume?
- On a 1 : 25 000 map a park covers 4 cm². What is its real area in hectares?
