Similar Figures Calculator
Calculate scale factors, missing measurements, and area/perimeter ratios for similar figures with step-by-step solutions.
Figure 1 (Original)
Figure 2 (Similar)
Understanding Similar Figures
What are Similar Figures?
Similar figures have the same shape but not necessarily the same size. They have equal corresponding angles and proportional corresponding sides. Similar figures can be obtained through dilation (scaling).
Properties of Similar Figures
Corresponding angles: Equal in measure
Corresponding sides: Proportional (same ratio)
Scale factor (k): Ratio of corresponding sides
Perimeter ratio: Equal to scale factor (k)
Area ratio: Equal to (scale factor)² or k²
Volume ratio: Equal to (scale factor)³ or k³ (for 3D figures)
Scale Factor Formula
Scale Factor: k = (corresponding side of Figure 2) ÷ (corresponding side of Figure 1)
Finding Missing Sides:
Missing side = (known corresponding side) × k
Example:
If sides of Figure 1 are 3, 4, 5 and one side of Figure 2 is 6:
k = 6 ÷ 3 = 2
Other sides of Figure 2: 4 × 2 = 8 and 5 × 2 = 10
Area and Perimeter Relationships
If scale factor = k, then:
• Perimeter ratio = k : 1
• Area ratio = k² : 1
Example: If k = 3
• Perimeter is 3 times larger
• Area is 3² = 9 times larger
Triangle Similarity Theorems
- AA (Angle-Angle): If two angles are equal, triangles are similar
- SSS (Side-Side-Side): If all three sides are proportional, triangles are similar
- SAS (Side-Angle-Side): If two sides are proportional and included angle is equal, triangles are similar
Real-World Applications
- Maps and Models: Scale drawings and architectural models
- Photography: Image resizing and cropping
- Navigation: Calculating distances using map scales
- Engineering: Creating scaled prototypes
- Shadow Problems: Finding heights using similar triangles
- Optics: Lens magnification and image formation
Frequently Asked Questions
What's the difference between similar and congruent figures?
Congruent figures have the same shape AND size (scale factor = 1). Similar figures have the same shape but can be different sizes (any scale factor). All congruent figures are similar, but not all similar figures are congruent.
How do you find the scale factor between two similar figures?
Divide any side of the second figure by its corresponding side in the first figure: k = side₂ ÷ side₁. For example, if corresponding sides are 6 and 9, the scale factor is 9 ÷ 6 = 1.5.
Why is the area ratio the square of the scale factor?
Area is 2-dimensional, so it's affected by the scale factor twice (length × width). If each dimension is multiplied by k, the area is multiplied by k × k = k². For example, if k = 2, area increases by 2² = 4 times.
Can all rectangles be similar to each other?
No! Rectangles are only similar if their corresponding sides are proportional. A 2×4 rectangle is similar to a 3×6 rectangle (ratio 1:2 for both), but not to a 2×5 rectangle (different ratios).
How do you prove two triangles are similar?
Use one of three methods: (1) AA - show two angles are equal, (2) SSS - show all three sides are proportional, or (3) SAS - show two sides are proportional and the included angle is equal.
If the perimeter doubles, does the area double?
No! If the perimeter doubles, the scale factor is 2, so the area quadruples (2² = 4). This is a common mistake. Perimeter increases linearly with scale factor, but area increases with the square of the scale factor.
How are similar figures used in real life?
Similar figures are everywhere: map scales (1 inch = 1 mile), architectural blueprints, photo enlargements, model cars/buildings, shadow calculations, and measuring heights indirectly using similar triangles.
Are all circles similar?
Yes! All circles are similar to each other because they all have the same shape. The scale factor between two circles equals the ratio of their radii (or diameters). This is why π is constant for all circles.