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SPI Mathematics · Triangles

Areas of Triangles and Similar Triangles

Triangles with the same height have areas in the ratio of their bases, and similar triangles have areas in the ratio of the squares of their corresponding sides.

All Triangles notes

Why this matters

Four questions in the bank, three EASY and one MODERATE. Two ask for an area ratio from a side or perimeter ratio, one goes back from areas to heights, and one compares two triangles that share a vertex. The usual slip is forgetting to square, or to take the square root on the way back.

Concept 1 of 2: Triangles with the same height

The area of a triangle is half its base times its height. When two triangles share a corner and their bases lie on one line, they have the same height, so their areas compare exactly as their bases do.

Definition

  • Area of a triangle =12×base×height= \frac{1}{2} \times \text{base} \times \text{height}.
  • Two triangles with the same height have areas in the ratio of their bases.
  • A median (a line from a corner to the midpoint of the opposite side) cuts a triangle into two triangles of equal area: equal bases, same height.

Area of a triangle

Area=12×base×heightar1ar2=base1base2 (same height)\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \qquad \frac{\text{ar}_1}{\text{ar}_2} = \frac{\text{base}_1}{\text{base}_2} \text{ (same height)}

Worked example

D is a point on side BC of △ABC\triangle ABC with BD = 6 cm and DC = 9 cm. Find the ratio of the areas of △ABD\triangle ABD and △ADC\triangle ADC.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Triangles · Areas of Triangles and Similar Triangles

In fig RP : PK = 3 : 2 then Area of △\triangle TRP : Area of △\triangle TPK =

Squaring a ratio that should not be squared

Triangles that only share a height are not similar, so their areas go as the bases, not as the squares of the bases. Bases 6 and 9 give areas 2 : 3, not 4 : 9.

Concept 2 of 2: Areas of similar triangles

Double every side of a triangle and both its base and its height double, so its area grows four times. Areas of similar triangles grow as the SQUARE of the scale factor.

Definition

  • If two triangles are similar with sides in the ratio a:ba : b, their areas are in the ratio a2:b2a^2 : b^2.
  • The same holds with corresponding heights, medians or perimeters in place of sides, since they all grow by the same factor.
  • Going back from areas to sides (or heights), take the square root of the area ratio.

Area ratio of similar triangles

ar(△ABC)ar(△DEF)=(ABDE)2=(height1height2)2\frac{\text{ar}(\triangle ABC)}{\text{ar}(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{\text{height}_1}{\text{height}_2}\right)^2

Worked example

Two similar triangles have corresponding sides 3 cm and 5 cm. The smaller has area 18 cm2\text{cm}^2. Find the area of the larger.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Triangles · Areas of Triangles and Similar Triangles

The Areas of two similar triangles are 81 cm2\text{cm}^2 and 49 cm2\text{cm}^2 respectively. Then the ratio of their corresponding heights will be

Giving the side ratio as the area ratio

Sides in the ratio 2 : 3 give areas 4 : 9. Leaving the ratio as 2 : 3 is the most common wrong option.

Forgetting the square root on the way back

Areas 64 and 121 give sides (and heights) 8 : 11. Writing 64 : 121 as the side ratio skips the square root.

Stopping at the whole triangle

When a median splits the similar triangle, find the whole triangle's area first, then halve it. The whole area is usually printed as a wrong option.
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Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Triangles with the same height

    Area of a triangle

    Area=12×base×heightar1ar2=base1base2 (same height)\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \qquad \frac{\text{ar}_1}{\text{ar}_2} = \frac{\text{base}_1}{\text{base}_2} \text{ (same height)}
  • Areas of similar triangles

    Area ratio of similar triangles

    ar(△ABC)ar(△DEF)=(ABDE)2=(height1height2)2\frac{\text{ar}(\triangle ABC)}{\text{ar}(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{\text{height}_1}{\text{height}_2}\right)^2

Watch out for (4)

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