A dilation multiplies every distance from a fixed centre by the same scale factor.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Not a rigid motion
Unless the factor is 1, lengths change. A dilation is the first transformation in these courses that is not rigid.
Angles survive
Angles are unchanged by a dilation, which is exactly why the shape is preserved while the size is not.
Reading the factor
A factor above 1 enlarges and one between 0 and 1 shrinks. A factor of 1 changes nothing.
Area scales by the square
Doubling every length multiplies area by 4, not 2. This is the squared relationship that ties the chapter to quadratics.
And volume by the cube
Doubling every length multiplies volume by 8. Each extra dimension adds another factor.
Changing size without changing shape
A dilation scales every distance from a fixed centre by the same factor. Angles are preserved and lengths are not, which is exactly the transformation congruence excluded.
What the factor does
Greater than 1 enlarges, between 0 and 1 shrinks, exactly 1 changes nothing. The centre is the one point that never moves, whatever the factor.
Images stay parallel
A dilated segment is parallel to the original, because the direction from the centre is unchanged. That parallelism is what drives the side-splitter theorem and most similarity proofs.
Area scales by the square
A factor of 3 multiplies lengths by 3 and areas by 9. Forgetting to square is the most common error in similarity problems that involve area rather than length.
Step 2: Try It Yourself
Tap and try it out.
Every length is multiplied by 2, but every angle is unchanged. The two figures are the same shape at a different size, which is what similar means.
Step 3: Watch an Example
One step at a time.
Watch Kofi Scale a Rectangle
A 3 by 4 rectangle is dilated by a factor of 3.
- Step 1
Every length multiplies by 3, so the sides become 9 and 12.
Step 4: Your Turn
Practice makes it stick.
The Enlargement
Problem 1 of 2
A length of 5 is dilated by a factor of 3. What is the new length?
The Area
Problem 2 of 2
An area of 12 is dilated by a factor of 3. What is the new area?
Scale It
1 of 8
Length 8 dilated by factor 2. New length?
2 of 8
Area 20 dilated by factor 2. New area?
3 of 8
Volume 5 dilated by factor 2. New volume?
4 of 8
An angle of 50° dilated by factor 3. What is the new angle, in degrees?
5 of 8
Length 12 dilated by factor 0.5. New length?
6 of 8
Area 36 dilated by factor 0.5. New area?
7 of 8
Sort each quantity by whether a dilation changes it.
Tap something to move it.
- Empty
- Empty
8 of 8
Area 7 dilated by factor 4. New area?
Step 5: Quick Check
Show what you know.
Question 1 of 2
An area of 10 is dilated by a factor of 3. What is the new area?
Question 2 of 2
Why does area scale by the square of the factor?
What You Learned
- A dilation multiplies every distance from a centre by the scale factor.
- Angles are unchanged, so the shape is preserved.
- Area scales by k² and volume by k³.