Quick Answer: Ratio of Area of Similar Triangles is a maths lab model for the theorem that the ratio of the areas of two similar triangles equals the square of the ratio of their corresponding sides. Filling a larger triangle with small congruent copies of a smaller one lets students count the area ratio instead of taking it on trust.
Why Areas Grow with the Square
If the sides of a triangle are doubled, its area does not double; it becomes four times as large. Dividing each side of a triangle into two equal parts and joining the points of division splits it into four congruent triangles, each similar to the whole with sides half as long. Dividing each side into three parts gives nine small triangles, and into four parts sixteen. The pattern k² for a scale factor k appears on the model before it appears in a formula.
In symbols, if triangle ABC is similar to triangle PQR, then area(ABC) : area(PQR) = AB² : PQ² = BC² : QR² = CA² : RP². The theorem is proved in secondary geometry by writing each area as ½ × base × height and using the fact that corresponding heights are in the same ratio as corresponding sides. Watching small triangles fill the large one gives that proof a clear picture.
The model also links to the mid-point theorem, since the triangle formed by joining the mid-points of the sides has sides half as long and exactly one quarter of the area. How the small triangles are held in place, and which scale factors are shown, should be confirmed with LabEquip.
Specifications
| Item | Model of the area ratio of similar triangles |
| Theorem | Area ratio equals the square of the ratio of corresponding sides |
| Method | Congruent small triangles filling a larger similar triangle |
| Typical level | Secondary geometry (similarity) |
| Scale factors shown and construction | Confirm at enquiry |
Applications
- Demonstrating that a scale factor of 2 gives four times the area and 3 gives nine times
- Supporting the formal proof that uses base and height
- Connecting the theorem to the mid-point theorem
- Solving problems in which one area and a side ratio are given
Care & Handling
- Keep all the small triangles in their tray, because one missing piece hides the pattern.
- Wipe with a dry cloth, and keep the pieces flat and away from heat.
- Count the pieces with the class at the end of the lesson as a quick recap of k².
Why Choose LabEquip
Teachers covering similarity in secondary classes use this model alongside a proof model such as the Pythagoras Theorem, Senior, since both rest on similar triangles. LabEquip lists them in its Mathematics School Lab Products range; ask about class quantities on the contact page.
Frequently Asked Questions
What does the theorem on areas of similar triangles state?
If two triangles are similar, the ratio of their areas equals the square of the ratio of any pair of corresponding sides. So if the sides are in the ratio 2 : 3, the areas are in the ratio 4 : 9.
Why do areas scale by the square of the scale factor?
Area depends on two lengths, a base and a height, and both are multiplied by the scale factor k. Multiplying two lengths each by k multiplies their product by k². The small-triangle model shows this as 4, 9 or 16 pieces filling triangles with sides 2, 3 or 4 times as long.
What is the ratio of the areas if the sides are in the ratio 1 : 3?
The areas are in the ratio 1 : 9. A triangle with sides three times as long as a smaller similar triangle can be filled exactly by nine copies of the smaller one.
Does the same rule apply to other shapes?
Yes. For any pair of similar plane figures, including squares, circles and polygons, the areas are in the ratio of the squares of corresponding lengths. For similar solids, volumes go a step further and scale with the cube.
How is the theorem connected to the mid-point theorem?
Joining the mid-points of the three sides of a triangle creates four congruent triangles, each similar to the original with sides half as long. Each small triangle therefore has one quarter of the area, which agrees with squaring the side ratio of one half.
Which class studies this theorem?
It belongs to the chapter on similar triangles in secondary classes. Even where the formal proof is not examined, students are expected to apply the result, and the model makes the reason for it clear.
Last Updated: September 2026
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