Practice Maths

Enlargement and Scale Factor

Key Ideas

Key Terms

Enlargement
A transformation that changes the size of a shape while keeping its shape exactly the same; described by a scale factor k and a centre of enlargement.
Scale factor
The ratio k = image length ÷ original length; if k > 1 the image is larger, if 0 < k < 1 the image is smaller (a reduction).
Similar figures
Figures with the same shape but (possibly) different sizes — all corresponding angles are equal and all corresponding sides are in the same ratio.
Centre of enlargement
The fixed point from which all measurements are scaled; every point moves k times as far from this centre in the same direction.
Corresponding sides
Sides in the same position in similar figures; their lengths are in the same ratio (the scale factor).
Ratio
A comparison of two quantities by division, used to express scale factors and relationships between corresponding lengths.

Finding the Scale Factor

If a side of length 4 cm maps to a side of length 10 cm: scale factor = 10 ÷ 4 = 2.5

Hot Tip Always use corresponding sides when calculating a scale factor — sides that are in the same position on both shapes. Check that all ratios are equal to confirm similarity.

Worked Example

Question: Triangle ABC has sides 3 cm, 4 cm and 5 cm. Triangle DEF is similar with DE = 9 cm (corresponding to AB = 3 cm). Find the other two sides of triangle DEF.

Step 1 — Find the scale factor.
k = DE ÷ AB = 9 ÷ 3 = 3

Step 2 — Multiply each corresponding side by k.
EF = 4 × 3 = 12 cm
DF = 5 × 3 = 15 cm

Triangle DEF has sides 9 cm, 12 cm and 15 cm.

What Is Enlargement?

An enlargement is a transformation that changes the size of a figure while keeping its shape exactly the same. The result is a figure that is similar to the original — same angles, but different side lengths. The amount of scaling is described by the scale factor k.

If k > 1, the image is larger than the original (an enlargement). If 0 < k < 1, the image is smaller than the original (a reduction). If k = 1, the image is the same size — no change. The image is always congruent to the original only when k = 1.

Every enlargement has a centre of enlargement — a fixed point from which all measurements are scaled. Points move away from (or toward) the centre by the scale factor.

Enlargement Diagram — Scale Factor 2

The diagram below shows two similar rectangles. The smaller rectangle (pre-image) has dimensions 3 × 2 and the larger rectangle (image) has dimensions 6 × 4 — every side has been multiplied by a scale factor of k = 2.

3 units 2 units Pre-image k = 2 6 units 4 units Image Each side × 2  |  Area × 4  (= k² = 2²)

Image Lengths and Area

When a figure is enlarged by scale factor k:

  • Each length in the image = k × the corresponding original length.
  • The area of the image = k2 × the original area. (Because area involves two dimensions, both of which are multiplied by k.)
  • The volume of a similar 3D solid = k3 × the original volume. (Three dimensions, each multiplied by k.)

Example: A rectangle is 5 cm × 3 cm. Area = 15 cm2. After enlargement with k = 3: new dimensions = 15 cm × 9 cm. New area = 135 cm2 = 32 × 15 = 9 × 15 = 135. ✓

This explains why doubling all dimensions of a box makes it 8 times heavier (not twice as heavy): volume scales by k3 = 23 = 8.

Similarity vs Congruence

Two figures are similar (~) if they have the same shape but (possibly) different sizes. All angles are equal and all sides are in the same ratio (the scale factor). Two figures are congruent (≅) if they are the same shape AND the same size — congruence is a special case of similarity with k = 1.

An enlargement always produces a similar figure. It produces a congruent figure only when k = 1 (no change in size).

When naming similar figures, list corresponding vertices in the same order, just as you do for congruent figures. The order tells you which angles and sides correspond.

Real-World Applications

Scale factors and enlargements appear everywhere in real life: maps (a map scale of 1:50 000 means 1 cm on the map = 50 000 cm = 500 m in real life); blueprints and floor plans (architects draw buildings at a reduced scale); photography and cinema (images enlarged for print or screen); models and toys (a model car at scale 1:18 is 18 times smaller than the real car).

To find a real-world length from a map: real length = map length × scale denominator. To find a map length from a real length: map length = real length ÷ scale denominator.

Key tip: The most common mistake with enlargement is applying the scale factor incorrectly to area and volume. Remember: lengths multiply by k, areas multiply by k2, and volumes multiply by k3. If a model car is 110 the length of the real car (k = 110), its volume is (110)3 = 11000 of the real car’s volume — a huge difference. Never just multiply the area or volume by k directly.

Mastery Practice

  1. Calculate the scale factor for each enlargement. Fluency

    1. Original side: 5 cm, Image side: 15 cm.
    2. Original side: 8 cm, Image side: 4 cm.
    3. Original side: 6 cm, Image side: 9 cm.
    4. Original side: 12 cm, Image side: 3 cm.
    5. Original side: 7 cm, Image side: 21 cm.
    6. Original side: 20 cm, Image side: 5 cm.
    7. Original side: 9 cm, Image side: 6 cm.
    8. Original side: 4 cm, Image side: 10 cm.

  2. Find the unknown side length. All pairs of shapes are similar. Fluency

    1. Scale factor 3. Original side: 7 cm. Image side: ?
    2. Scale factor 0.5. Original side: 14 cm. Image side: ?
    3. Similar triangles. One side of original: 5 cm, corresponding image side: 15 cm. Another original side: 8 cm. Find the corresponding image side.
    4. Similar rectangles. Original is 4 cm × 9 cm. Image width: 12 cm. Find the image length.
    5. Similar triangles. Original sides: 6, 8, 10 cm. Image shortest side: 9 cm. Find the other two image sides.
    6. Scale factor ½. Original triangle sides: 14 cm, 20 cm, 24 cm. Find all image sides.

  3. State whether each pair of figures is similar, congruent, or neither. Give a reason. Understanding

    1. Triangle with sides 3, 4, 5 cm and triangle with sides 6, 8, 10 cm.
    2. Rectangle 5 cm × 8 cm and rectangle 10 cm × 16 cm.
    3. Rectangle 3 cm × 6 cm and rectangle 4 cm × 8 cm.
    4. Square with side 5 cm and square with side 7 cm.
    5. Triangle with sides 4, 7, 9 cm and triangle with sides 8, 12, 16 cm.
    6. Two equilateral triangles, one with side 3 cm and one with side 9 cm.
    7. Triangle with angles 30°, 60°, 90° and sides 5, 8.66, 10 cm, and triangle with angles 30°, 60°, 90° and sides 3, 5.2, 6 cm.

  4. Use similar figures to find the unknown measurements. Understanding

    1. Two similar triangles: Triangle A has sides 5 cm, 12 cm and x cm. Triangle B has sides 10 cm, 24 cm and 26 cm. Find x.
    2. Two similar pentagons. One side of shape A is 8 cm; the corresponding side of shape B is 20 cm. Another side of shape A is 6 cm. Find the corresponding side of shape B.
    3. A photograph is 10 cm × 15 cm. It is enlarged so the new width is 25 cm. What is the new length?
    4. A model car is built to a scale of 1:24. The real car is 4.8 m long. How long is the model car in cm?
    5. Two similar triangles have perimeters of 30 cm and 45 cm. The scale factor is __? A side of the smaller triangle is 8 cm. Find the corresponding side of the larger triangle.

  5. Problem solving with enlargements and scale factors. Problem Solving

    1. A builder uses a scale drawing where 1 cm represents 2.5 m on the scale 1:250.
      1. A wall is 8.5 m long. How long is it on the drawing (in cm)?
      2. A room on the drawing measures 4.2 cm × 3.6 cm. What are the real dimensions?
    2. A tree casts a shadow 9 m long. At the same time, a 1.5 m tall person casts a shadow 2.25 m long. Use similar triangles to find the height of the tree.
    3. A photographer wants to enlarge a photo from 6 cm × 9 cm. The largest available paper is 20 cm wide.
      1. What is the maximum scale factor to fit the width?
      2. What will the length of the enlarged photo be?
      3. Will the 20 cm paper be long enough? Explain.
    4. A map has a scale of 1:50 000. Two towns are 8.4 cm apart on the map.
      1. What is the real distance between the towns in km?
      2. A road on the map is 12.5 cm long. What is the real length in km?

  6. Effect of scale factor on perimeter and area. Problem Solving

    Key fact. When a shape is enlarged by scale factor k, its perimeter is multiplied by k and its area is multiplied by k².
    1. A rectangle has perimeter 24 cm and area 32 cm². It is enlarged by scale factor 3. Find the new perimeter and the new area.
    2. A triangle has area 20 cm². After enlargement, the area is 180 cm². What is the scale factor?
    3. A square has side length 5 cm and is enlarged with scale factor 2.5. Find the new side length, the new perimeter, and the new area.
    4. A shape has area 48 m². It is reduced by scale factor 0.5. What is the area of the reduced shape?
    5. Two similar rectangles have areas 25 cm² and 100 cm². Find the scale factor from the smaller to the larger rectangle.

  7. Enlargements on the coordinate plane. Problem Solving

    Centre of enlargement. To enlarge a point from a centre C by scale factor k, move each point k times as far from C in the same direction. If C = (0, 0), multiply each coordinate by k.
    1. Triangle A has vertices (1, 1), (3, 1), (1, 4). It is enlarged from the origin with scale factor 2. Write the coordinates of the image triangle.
    2. A rectangle has vertices (0, 0), (4, 0), (4, 2), (0, 2). It is enlarged from the origin by scale factor 1.5. Find the image vertices.
    3. Point P = (6, 9) is the image of point Q under an enlargement from the origin with scale factor 3. Find the coordinates of Q.
    4. Triangle A has vertices (2, 2), (6, 2), (2, 6). It is enlarged from centre (0, 0) with scale factor 0.5. Find the image vertices and state whether the image is an enlargement or a reduction.
    5. A shape is enlarged from the origin. The original point is (3, 4) and the image is (9, 12). What is the scale factor?

  8. Multi-step scale factor problems. Problem Solving

    Strategy. Identify the scale factor first, then apply it to find all unknowns. For area and perimeter, remember the multiplier rules.
    1. A model aeroplane is built to a scale of 1:72. The real wingspan is 36 m. Find the model wingspan in cm.
    2. A floor plan is drawn at scale 1:200. On the plan, a room is 3.5 cm × 4 cm.
      1. Find the real dimensions of the room in metres.
      2. Find the real area of the room.
      3. What is the area on the plan? How many times larger is the real area than the plan area?
    3. Two similar triangles have corresponding sides of 8 cm and 20 cm.
      1. Find the scale factor from small to large.
      2. The smaller triangle has perimeter 28 cm. Find the perimeter of the larger.
      3. The smaller triangle has area 24 cm². Find the area of the larger.

  9. Reasoning about similarity and scale. Problem Solving

    Real-world similarity. Many real-world problems use similar triangles or scale factors to find lengths that cannot be measured directly.
    1. A flagpole casts a 6 m shadow. At the same time, a 1.8 m fence post casts a 0.9 m shadow. Use similar triangles to find the height of the flagpole.
    2. A photograph 8 cm wide and 12 cm long is to be enlarged so that the length becomes 30 cm. Will the enlarged photo fit on a sheet of paper 20 cm wide? Show all working.
    3. Two similar pentagons have perimeters of 45 cm and 75 cm.
      1. Find the scale factor from the smaller to the larger pentagon.
      2. One side of the smaller pentagon is 9 cm. Find the corresponding side of the larger.
      3. The smaller pentagon has area 81 cm². Find the area of the larger pentagon.
    4. A painting is 60 cm wide and 45 cm tall. It is to be reproduced as a greeting card 10 cm wide.
      1. What is the scale factor from painting to card?
      2. What will be the height of the card image?
      3. What fraction of the original area does the card image have?

  10. Investigate the effect of successive enlargements. Problem Solving

    Successive enlargements. Applying two scale factors in succession gives a combined scale factor equal to their product.
    1. A shape is enlarged first by scale factor 2, then the image is enlarged again by scale factor 3. What is the overall scale factor from the original to the final image?
    2. A side of 5 cm is enlarged by scale factor 2, and then that image is enlarged by scale factor 1.5.
      1. Find the length after the first enlargement.
      2. Find the length after the second enlargement.
      3. Confirm this equals the original × the combined scale factor.
    3. A square with side 4 cm is enlarged by scale factor 2.5. Then the image is reduced by scale factor 0.4.
      1. What is the final side length?
      2. What is the overall scale factor?
      3. Explain what has happened to the shape overall.
    4. A photo is 6 cm × 9 cm. It is enlarged to A4 (21 cm wide). It is then reduced back by scale factor ½. What are the final dimensions?
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