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Home›Geometry and drawings›Triangles

Similar triangles and scale

A 1.6 m student casts a 2 m shadow while the school flagpole casts one of 15 m. The pole is 1.6 × 15 ÷ 2 = 12 m tall, and nobody had to climb it.

How Thales measured a pyramid from the ground

BackgroundThales of Miletus lived in the sixth century BC. OpenStax Contemporary Mathematics tells the story that he found the height of an Egyptian pyramid without climbing it. He waited for the moment when the tip of his own shadow reached the tip of the pyramid’s shadow. Then the sun’s rays, his body and his shadow made a small right triangle, and the same rays with the pyramid and its shadow made a large one. The two triangles had the same angles, so their sides were in the same proportion, and his own height and two shadow lengths were enough to work out the pyramid’s height. The same book credits Thales with noticing that the base angles of an isosceles triangle are equal, which you proved in the last unit.

You can repeat the trick on a sunny day at school. A student 1.6 m tall casts a 2 m shadow. At the same moment the flagpole casts a 15 m shadow. The sun’s rays arrive at the same angle everywhere in the playground, so the student and the pole each form a right triangle with the same angles. Height divided by shadow must be the same for both: 1.6 ÷ 2 = 0.8, so the pole is 15 × 0.8 = 12 m. OpenStax does the same sum with a radio tower: a 9 ft truck with a 15 ft shadow means a tower with a 180 ft shadow is 180 × 9 ÷ 15 = 108 ft tall.

The student’s triangle and the pole’s triangle have the same angles, so height and shadow are in the same ratio, 1.6 : 2 = 12 : 15. The two are drawn at different scales.
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Shadow method. Move the sun slider: both shadows change, but height ÷ shadow stays equal for the stick and the tower, and the panel recovers the tower’s height from the stick.

What similar means

Two figures are similar when one is an enlargement or reduction of the other. OpenStax Prealgebra puts it simply: same shape, different size, one a scale model of the other. For triangles that means two things at once. Matching angles are equal. And matching sides are all in the same ratio, called the scale factor. If triangle ABC is similar to triangle DEF with scale factor 2, then DE = 2 × AB, EF = 2 × BC and FD = 2 × CA, while the angles at D, E and F are exactly those at A, B and C. In Korean books this is written △ABC ∽ △DEF, and the ratio of matching sides, here 1 : 2, is called the similarity ratio.

How do you decide that two triangles are similar without measuring everything? Euclid’s Proposition 4 of Book VI says that triangles with equal angles have proportional sides, so equal angles are enough, and because the angle sum is 180°, two equal angles already give the third. That is the angle-angle test, the one Thales used. Two more tests mirror the congruence conditions: all three pairs of sides in the same ratio, or two pairs of sides in the same ratio with the included angle equal. Only the angle-angle test has no length in it, which is why it is the one you reach for with shadows.

Solving for a missing side with a proportion

Once you know two triangles are similar, write one fraction for each pair of matching sides and set them equal. The danger is matching the wrong sides. A safe habit is to pair sides by the angles they face: the side opposite the right angle in one triangle goes with the side opposite the right angle in the other, and so on. Suppose △ABC ∽ △DEF, AB = 4, BC = 6, DE = 10. Then DE : AB = 10 : 4 = 2.5, so EF = 6 × 2.5 = 15. Check with the other pair if you know it; every matching pair must give the same 2.5.

Areas grow with the square of the scale

A scale factor of 2 doubles every length, but it does not double the area. The base doubles and the height doubles, so base × height ÷ 2 grows 2 × 2 = 4 times. With scale factor k, areas multiply by k². A photo printed at three times the width needs nine times as much paper. Perimeters, being lengths, grow only k times: similar triangles with a similarity ratio of 2 : 3 and a smaller perimeter of 30 cm have a larger perimeter of 45 cm.

  1. A 1.6 m student casts a 2 m shadow. At the same time a flagpole casts a 15 m shadow. How tall is the pole?
  2. A 9 ft truck casts a 15 ft shadow while a radio tower casts a 180 ft shadow. How tall is the tower?
  3. A 5 ft person casts a 7 ft shadow, and a tree casts a 57 ft shadow. How tall is the tree, to one decimal place?
  4. △ABC ∽ △DEF, AB = 4, BC = 6 and DE = 10. Find EF.
  5. Two triangles are similar with scale factor 3. The smaller has area 5 cm². What is the area of the larger?
  6. Similar triangles have similarity ratio 2 : 3. The smaller perimeter is 30 cm. Find the larger perimeter.
  7. Triangle 1 has angles 50° and 70°. Triangle 2 has angles 70° and 60°. Are they similar?
    • Yes
    • No
    • Only if sides are given
    • Only if both are right triangles
  8. Sides 3, 4, 5 and sides 6, 8, 11. Are the triangles similar?
    • No
    • Yes, scale factor 2
    • Yes, scale factor 2.2
    • Only the angles are equal
  9. In OpenStax, matching sides 4 and 7 in one triangle correspond to 2 and 3.5 in the other. The scale factor from the first to the second is:
    • 0.5
    • 2
    • 0.57
    • 1.75
  10. On a map at scale 1 : 50 000, two towns are 6 cm apart. How far apart are they in kilometres?
Common mistakes
  • “The scale factor is 2, so the area doubles.” Both the base and the height double, so the area is 2 × 2 = 4 times as big. Lengths scale by k, areas by k².
  • “△ABC ∽ △DEF with AB = 4 and EF = 10, so the scale factor is 2.5.” AB matches DE, not EF. Pair sides by the order of the letters, or by the angles they face, before dividing.
  • “Add the same amount to every side and the triangle stays similar.” Adding 2 to sides 3, 4, 5 gives 5, 6, 7, and the ratios 5 ÷ 3, 6 ÷ 4, 7 ÷ 5 all differ. Similarity needs multiplying, not adding.
Teacher's noteThe 2022 Korean curriculum places similarity in the middle school band, with one standard for the meaning of similarity and the similarity ratio and another for the triangle similarity conditions. Its term list includes the similarity ratio and the sign ∽. The curriculum also recommends teaching tools and software for the meaning of congruence and similarity. The shadow measurement works well as an outdoor task: in groups, measure one person and one shadow, then a tree or a flagpole’s shadow, and compare answers between groups. Different groups get slightly different heights, which opens a useful talk about measurement error.
Key points
  • Similar triangles: equal matching angles and matching sides in one ratio.
  • Tests: SSS ratio, SAS ratio with included angle, or two equal angles (AA).
  • Pair sides by the order of letters or by the angles they face.
  • Scale factor k: lengths and perimeters times k, areas times k².
Sources for this course (7)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수03-08] 이등변삼각형·정삼각형, [4수03-09] 직각삼각형·예각삼각형·둔각삼각형, [4수03-25] 삼각형과 사각형의 내각의 크기의 합을 여러 가지 방법으로 추론; 초등학교 5~6학년군 [6수03-01] 합동·대응점·대응변·대응각, [6수03-14] 삼각형의 넓이(높이가 삼각형의 외부에 있는 것도 다룸); 중학교 1~3학년군 [9수03-03] 삼각형의 작도, [9수03-04] 삼각형의 합동 조건, [9수03-05] 삼각형의 내각과 외각, [9수03-09] 이등변삼각형의 성질을 합동 조건으로 정당화하고 증명이라는 용어 도입, [9수03-12]·[9수03-13] 닮음·닮음비·삼각형의 닮음 조건, [9수03-15] 피타고라스 정리를 추측하고 정당화, 역은 직관적으로 이해하고 직각삼각형 판별에 사용; 용어와 기호 대변, 대각, 내각, 외각, 증명, 닮음비, △ABC, ≡, ∽)
  • Textbook: OpenStax, Contemporary Mathematics, 10.3 Triangles (triangle from three noncollinear points; acute, right, obtuse; side a opposite angle A; hypotenuse and legs; angle sum 180° by a parallel line and alternate interior angles; x, x + 17, 3x − 62 gives 45°, 62°, 73°; SSS, SAS, ASA, AAS; similar triangles, scaling factor; tower shadow 180 ft with a 9 ft truck and 15 ft shadow gives 108 ft; Thales and the pyramid: openstax.org/books/contemporary-mathematics/pages/10-3-triangles)
  • OpenStax Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; Pythagoras around 500 BCE; 13 ft ladder 5 ft from the wall; 10 inch bracket about 7.1 inches; similar triangles: openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem)
  • 9.4 Use Properties of Rectangles, Triangles, and Trapezoids (A = ½bh from half a rectangle; isosceles and equilateral; equilateral perimeter 93 in: .../pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids)
  • OpenStax Precalculus 2e, 8.2 Non-right Triangles: Law of Cosines (Heron’s formula with semi-perimeter s; sides 10, 15, 7 give s = 16 and area about 29.4; Chicago lot 62.4, 43.5, 34.1 m gives about 711.4 m²: openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines)
  • History for story blocks only: Euclid’s Elements, D. E. Joyce edition, Clark University (Book I Prop. 20 triangle inequality and Heron’s shortest-path problem; Prop. 26 and its guide on congruence theorems, SSA ambiguity and side-side-right angle; Prop. 32 exterior angle and angle sum, Proclus corollaries; Prop. 47 and its guide on Old Babylonians ca. 1900 to 1600 BCE, Plimpton 322, the Zhou bi hypotenuse diagram, Proclus; Prop. 48 converse; Book VI Prop. 4 equiangular triangles: mathcs.clarku.edu/~djoyce/elements/bookI/propI47.html)
  • MacTutor History of Mathematics, University of St Andrews, Heron of Alexandria (formula proved in Metrica Book I; eclipse of 13 March 62; side of 720 about 26 5/6: mathshistory.st-andrews.ac.uk/Biographies/Heron/) MacTutor, "Euclid of Alexandria" (born about 325 BC, died about 265 BC; dating for the Elements, about 2,300 years ago: mathshistory.st-andrews.ac.uk/Biographies/Euclid/, checked 2026-10-19).

Checked: 2026-10-19