Similar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.
In Spanish
figuras semejantes
Also called
same shape, similar shapes, similar polygons, similar triangles
Every side of the big triangle is 2 times the matching side of the small one, and the angles match, so they are similar. The 3 by 5 rectangle fails: its sides do not grow by one common factor.
Formal definition and notation
Two figures are similar if one can be mapped onto the other by a sequence of rotations, reflections, translations and dilations. For polygons this means all pairs of corresponding angles are congruent and all pairs of corresponding sides are proportional, a′a = b′b = c′c = k. The constant k is the scale factor; if k = 1 the figures are congruent.
Symbol
Read it as
Meaning
∼
is similar to
△ ABC ∼ △ DEF: the triangles are similar, with A matching D, B matching E, C matching F
k
k
the scale factor from one figure to the other
≅
is congruent to
same shape and same size (scale factor 1)
Key facts
Similar does not mean equal size. Equal size and shape is congruent; similar allows a different size.
Two triangles are similar if two pairs of angles are equal (the AA criterion).
All circles are similar, and all squares are similar. Not all rectangles are: a 2 by 4 and a 3 by 5 rectangle are not.
A dilation, possibly combined with slides, flips and turns, maps a figure to a similar figure.
Maps, models and scale drawings are similar to the real objects they show.
Proportion for similar figures
Proportion for similar figures
side in imagematching side in original = k
k
scale factor: the same for every pair of matching sides
perimeter
multiplies by k
area
multiplies by k2
volume
multiplies by k3 (for similar solids)
Match the sides in the same order as the vertices: if △ ABC ∼ △ DEF, then DEAB = EFBC = DFAC.
Try it: dilate a shape
Interactive
Choose a scale factor to dilate the triangle from the origin. The image keeps the same angles while every side is multiplied by the scale factor, so the two triangles are similar.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Similar Figures examples, step by step
Example 1: Find the scale factor
Warm-up
A triangle has sides 5, 7 and 9. A similar triangle has a shortest side of 15. Find the scale factor and the other two sides.
The shortest sides match: k = 155 = 3.
Multiply the other sides by 3: 7 × 3 = 21 and 9 × 3 = 27.
Check: 217 = 279 = 3.
Answer: The scale factor is 3, and the other sides are 21 and 27.
Example 2: Missing side with a proportion
Core
△ ABC ∼ △ DEF. AB = 8, BC = 12 and DE = 10. Find EF.
The order of letters tells which sides match: AB matches DE, and BC matches EF.
Set up the proportion: EFBC = DEAB, so EF12 = 108.
Solve: EF = 12 × 108 = 15.
Answer: EF = 15.
Example 3: Shadows and height
Test-style
At the same time of day, a 6-foot person casts a 4-foot shadow and a tree casts a 30-foot shadow. How tall is the tree?
The sun hits both at the same angle, so the person and shadow and the tree and shadow form similar right triangles.
Height over shadow is the same: h30 = 64.
Solve: h = 30 × 64 = 45.
Check the scale factor: 30 ÷ 4 = 7.5, and 6 × 7.5 = 45.
Answer: The tree is 45 feet tall.
Common mistakes
✗ MistakeSimilar figures must be the same size.
✓ InsteadSimilar does not mean equal size. That is congruent. Similar figures have the same shape, and their sizes can differ.
Similarity allows any scale factor; congruence is the special case where the scale factor is 1.
✗ MistakeA 3 by 5 rectangle is enlarged so 3 becomes 6, so 5 becomes 5 + 3 = 8.
✓ InsteadMultiply by the same factor: k = 2, so 5 becomes 10.
Similar figures grow by multiplying every side by one scale factor, not by adding the same amount.
✗ Mistake△ ABC ∼ △ DEF with AB = 8, BC = 12, DE = 10: 810 = EF12, so EF = 9.6.
✓ InsteadKeep matching sides in the same position: EF12 = 108, so EF = 15.
Each ratio must compare the same figure on top (new over old, or old over new) on both sides of the proportion.
✗ MistakeIf the sides double, the area doubles.
✓ InsteadThe area multiplies by 22 = 4.
Area depends on two lengths, and both are doubled.
Congruent figures have the same shape and the same size (scale factor 1). Similar figures have the same shape; the size may differ. Every pair of congruent figures is also similar.
The scale factor is the number that multiplies every length. Similar figures are the shapes that this number relates; the 3-4-5 and 6-8-10 triangles have scale factor 2.
Quick check: 3 questions
Question 1 of 3
Is a 4 by 10 rectangle similar to a 6 by 15 rectangle?
Show answer and explanation
Answer: C.64 = 1.5 and 1510 = 1.5: one common factor, and all angles are 90°, so they are similar. Size does not matter (A), similarity uses ratios, not differences (B), and not all rectangles are similar (D).
Question 2 of 3
Two similar triangles have a scale factor of 3. The small triangle has an area of 10 square cm. What is the area of the large one?
Show answer and explanation
Answer: B. Areas scale by k2 = 9: 10 × 9 = 90. Choice A scales the area like a length, C adds 3, and D cubes the area.
Question 3 of 3
Which must be true about two similar figures?
Show answer and explanation
Answer: D. Similar figures always have equal corresponding angles and proportional sides. Size, perimeter and area are equal only when the scale factor is 1, which makes the figures congruent.
Similar Figures: frequently asked questions
What is the difference between similar and congruent?
Congruent figures have the same shape and the same size: you could place one exactly on top of the other. Similar figures have the same shape but can be different sizes, like a photo and its enlargement. Congruent figures are similar figures with a scale factor of 1.
Are all squares similar? All rectangles?
All squares are similar: every angle is 90° and all four sides grow by the same factor. All circles and all equilateral triangles are similar too. Rectangles are not always similar: a 2 by 4 rectangle and a 3 by 5 rectangle have equal angles, but their side ratios, 3/2 and 5/4, are different.
How can you tell if two triangles are similar?
Check angles or sides. If two angles of one triangle equal two angles of the other (AA), the triangles are similar, because the third angles must match too. Or check that all three pairs of sides have the same ratio (SSS similarity), as in 3-4-5 and 6-8-10.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.G.A.1, CCSS 8.G.A.4, CCSS HSG-SRT.A.2
Cite this entry
Effortless Math Team. “Similar Figures.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/similar-figures/
Flashcard
TermSimilar Figures∼
DefinitionSame shape, sizes may differ: equal angles and proportional sides. Triangles 3-4-5 and 6-8-10 are similar (factor 2).
Scale FactorA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · Grade 7, Grade 8, Algebra 1RotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometryReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryTranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryDilationA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Geometry · Grade 8, GeometryPolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5Corresponding AnglesCorresponding angles are two angles in the same relative position at each place where a transversal crosses two lines, such as the top-left corner at both crossings. When the two lines are parallel, corresponding angles are equal. A transversal across two lines makes four pairs of corresponding angles.Geometry · Grade 8, GeometryCongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometrySideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Scale DrawingA scale drawing is a drawing of a real object in which every length is multiplied by the same number, so it has the same shape but a different size. Its scale, such as 1 cm = 2 m, links a length on the drawing to the actual length. Maps, floor plans and blueprints are scale drawings.Ratios and Proportions · Grade 7Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8ProportionA proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7, Grade 8AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1