In geometry, similar shapes are figures that have the same shape but not necessarily the same size. Two shapes are considered similar when all corresponding angles are equal and all corresponding sides are in proportion. This mathematical relationship creates a powerful concept used in everything from art and architecture to engineering and computer graphics.
Choose from rectangles, triangles, circles, or regular polygons. For polygons, you can specify the number of sides.
Select what you want to calculate:
Input the dimensions for your original shape and (depending on your calculation type) either the similar shape’s dimensions or the scale factor.
The calculator will display comprehensive results including visual representations, precise measurements, and comparison details.
Architects use similarity to create scale models and blueprints that accurately represent buildings at a reduced size. The proportional relationships ensure that the design will function correctly when built at full scale.
For example, if a model uses a scale factor of 1:50, this means every measurement on the model represents a measurement 50 times larger on the actual building. The similar shapes principle ensures that all proportions remain consistent.
Cartographers rely on similar shapes to create accurate map representations of geographical areas. When properly scaled, maps maintain proportional relationships between distances, allowing for accurate navigation and spatial understanding.
The scale on a map (e.g., 1:24,000 on a topographic map) indicates that one unit on the map equals 24,000 of the same units in real life. This scaling creates a similar shape relationship between the map and actual terrain.
Engineers use similarity principles when designing prototypes and scaling products for manufacturing. Understanding how dimensions, forces, and physical properties change with scaling is crucial for successful design.
When an engineer scales up a successful small prototype to full production size, they must account for how the scale factor affects strength, weight, and material requirements. Our calculator helps visualize these relationships.
Artists and photographers use similar shapes when resizing images while maintaining aspect ratios, creating proportionally accurate drawings, and when scaling artwork for different applications.
The “golden ratio” (approximately 1.618:1) is often used in compositions to create aesthetically pleasing proportions in similar shapes across different scales.
k = dimension of similar shape / corresponding dimension of original shape
The scale factor is the fundamental ratio that determines how much larger or smaller one shape is compared to another similar shape.
Area ratio = k² = (area of similar shape) / (area of original shape)
The areas of similar shapes have a ratio equal to the square of the scale factor. This is why doubling the dimensions quadruples the area.
Perimeter ratio = k = (perimeter of similar shape) / (perimeter of original shape)
The perimeters of similar shapes have a ratio equal to the scale factor. This linear relationship is simpler than the area relationship.
Volume ratio = k³ = (volume of similar shape) / (volume of original shape)
When working with three-dimensional similar objects, the volume ratio equals the cube of the scale factor. This explains why doubling dimensions creates an object with eight times the volume.
Problem: A rectangular garden plot measures 12 feet by 8 feet. You want to create a similar garden that’s twice as large in each dimension.
Solution:
This example confirms that when the scale factor is 2, the area increases by a factor of 4 (2²), while the perimeter increases by a factor of 2.
Problem: Triangle ABC has sides of length 3 cm, 4 cm, and 5 cm. Triangle DEF is similar to ABC with a scale factor of 1.5. Find the sides and area of triangle DEF.
Solution:
The area of the similar triangle scales by the square of the scale factor, as expected from the theory of similar shapes.
Understanding similar shapes forms a critical part of geometry education from middle school through college levels. These concepts build important spatial reasoning skills and connect to many other mathematical topics:
Teachers can use our Similar Shapes Calculator as a teaching tool to demonstrate how changing dimensions affects area and perimeter, helping students visualize these relationships dynamically.
Two shapes are similar if (1) all corresponding angles are equal and (2) all corresponding sides are proportional. For polygons, this means they have the same number of sides and the same interior angles, with sides that are in proportion. For circles, all circles are similar by definition since they all have the same shape regardless of size.
Congruent shapes are identical in both shape and size, meaning they match exactly in all dimensions. Similar shapes have the same shape but can be different sizes, with proportional corresponding sides. You can think of congruent shapes as a special case of similar shapes where the scale factor equals exactly 1.
The scale factor has different effects depending on the dimension you’re measuring. If the scale factor is k:
This is why doubling the dimensions of a shape results in 4 times the area and 8 times the volume.
Yes, all circles are similar to each other. Since every circle has exactly the same shape (a perfect round figure where all points on the perimeter are equidistant from the center), circles only differ in size. The scale factor between any two circles is equal to the ratio of their radii or diameters.
Similar shapes have numerous practical applications:
The mathematical principles of similar shapes help professionals in these fields create accurate scaled versions while maintaining proper proportions.
This Similar Shapes Calculator is provided for educational and informational purposes only. While we strive for accuracy in all calculations, results should be verified for critical applications. The calculator assumes perfect geometric shapes and does not account for real-world variations or imprecisions in measurements.
Last Updated: March 4, 2025 | Next Review: March 4, 2026