Congruency, Enlargement and Combined Transformations
How to Find a scale factor and image area
Use this to find the scale factor of an enlargement and to work out the area of the enlarged image.
Before you start
- Multiplying and dividing lengths to compare two shapes
- Squaring numbers, including fractions and negatives
- Reading a scale drawing or a coordinate grid
When to use it
Use this to find the scale factor of an enlargement and to work out the area of the enlarged image.
The steps
- Find one pair of corresponding lengths on the object and the image.
- Divide the image length by the object length to get the scale factor k.
- Check the sign: a negative k means the image is on the other side of the centre.
- For area, use the given rule: area of image = k² × area of object.
- State the centre of enlargement clearly if the question asks for a full description.
Worked example
Under an enlargement with centre O, a triangle with a base of 3 cm and an area of 8 cm² is mapped to an image whose base is 9 cm. Find the scale factor and the area of the image.
- Corresponding lengths: object base = 3 cm and image base = 9 cm.
- Scale factor k = image ÷ object = 9 ÷ 3 = 3.
- The image is on the same side of O as the object, so k is positive: k = 3.
- Area of image = k² × area of object = 3² × 8 = 9 × 8 = 72 cm².
- Full description: an enlargement of scale factor 3, centre O.
A second example, with a twist
The image is smaller and lies on the opposite side of the centre, giving a negative fractional scale factor. Under an enlargement with centre P, a rectangle with a longer side of 8 cm and an area of 40 cm² is mapped to an image whose corresponding side is 4 cm and which lies on the opposite side of P.
Find the scale factor and the area of the image.
- Corresponding lengths: object side = 8 cm and image side = 4 cm.
- Scale factor k = image ÷ object = 4 ÷ 8 = ½.
- The image is on the opposite side of P, so k is negative: k = −½.
- Area of image = k² × area of object = (−½)² × 40 = ¼ × 40 = 10 cm².
- Full description: an enlargement of scale factor −½, centre P.
Formulae you may need
Formula pages
- Sum of interior angles of a polygon
- Area of kite
- Area of trapezium
- Surface area of cylinder
- Surface area of cone
- Surface area of sphere
- Volume of prism
- Volume of cylinder
- Volume of cone
- Volume of sphere
- Volume of pyramid
- Scale factor
- Area of image
Practise this in a KBAT problem
Frequently asked questions
Why is the area multiplied by k² and not just by k?
Because area is two-dimensional. An enlargement stretches every length by k, and area depends on two lengths, such as base times height, so it grows by k × k = k².
When k = 3 the lengths triple but the area becomes nine times larger, not three times.
Does a negative scale factor change the area?
No. A negative k only puts the image on the opposite side of the centre; the area still uses k².
Squaring removes the minus sign, so (−½)² = ¼, exactly the same as (½)². Always give the area as a positive value.
How do I know which side of the centre the image is on?
Draw a straight line from the centre through an object point. If the image point lies on the same side as the object, k is positive; if it lands on the far side of the centre, k is negative.
A diagram in the question usually makes this clear.