Perimeter plus tiles needed to build it

3.MD.D.84.MD.A.33.OA.A.3 · adapt · grade 4

Archetype: Perimeter by Tracing Every Side · step in a 11-type progression

There is a right triangle whose two legs (the two sides that form the right angle) are each $3\text{ cm}$ long. To make a rectangle with a perimeter of $30\text{ cm}$ by joining copies of this right triangle side by side, how many right triangles are needed?

(Figure) A right triangle whose two legs forming the right angle (the base and the height) are each labeled $3\text{ cm}$ .

Show solution

Understand

We have a right triangle whose two perpendicular legs are each 3 cm. We join copies of this triangle side by side to build a rectangle whose perimeter is 30 cm, and we want to know how many triangles that takes.

Givens

Each right triangle has two legs of 3 cm meeting at the right angle.
Two such triangles joined along their hypotenuse form a 3 cm by 3 cm square.
The finished rectangle must have a perimeter of 30 cm.
The triangles are joined side by side in a single row.

Unknowns

The number of right triangles needed to build the rectangle.

Constraints

The rectangle is made only of these triangles (no gaps or overlaps).
Lengths come in multiples of 3 cm because the building square is 3 cm on a side.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#6 Guess and Check

Drawing the triangles lets us see that two of them make a 3 cm square, turning the shape question into a tidy tiling. We then split the task into subproblems (make a square, line squares in a row, match the perimeter) and check which row length gives 30 cm.

Execute

#1 Draw a Diagram 3.G.A.1

Put two copies of the right triangle together along their slanted sides (the hypotenuses). Because both legs are 3 cm, the two triangles snap into a square that is 3 cm wide and 3 cm tall.

2 \text{ triangles} = 1 \text{ square } (3\text{ cm} \times 3\text{ cm})

Grade 3 shape sense: two matching right triangles fit together along the long side to make a square.

#7 Identify Subproblems 3.OA.A.3

Join the 3 cm squares side by side in one row. If there are n squares, the rectangle is 3 cm tall and 3 times n cm long.

\text{length} = 3 \times n, \quad \text{height} = 3

Grade 3 multiplication: laying equal 3 cm squares end to end multiplies the length by the number of squares.

#6 Guess and Check 3.MD.D.8

The perimeter of a rectangle is twice the length plus twice the height. Set it equal to 30 cm and find n: 2 times (3n + 3) = 30, so 3n + 3 = 15, so 3n = 12, so n = 4. We get a 12 cm by 3 cm rectangle.

2\,(3n + 3) = 30 \;\Rightarrow\; 3n + 3 = 15 \;\Rightarrow\; n = 4

Grade 3 perimeter sense: try row lengths until the trip around the rectangle measures 30 cm; 4 squares makes a 12 by 3 rectangle (perimeter 30).

#7 Identify Subproblems 3.OA.A.3

Each of the 4 squares is made from 2 triangles, so the rectangle needs 4 times 2 = 8 triangles.

4 \times 2 = 8

Grade 3 multiplication: 4 squares, 2 triangles each, is 8 triangles.

Answer: 8 triangles

Review

The rectangle is 12 cm by 3 cm, and its perimeter is 12 + 3 + 12 + 3 = 30 cm, exactly as required. Eight right triangles with legs of 3 cm cover an area of 8 times (3 times 3 divided by 2) = 36 square cm, which equals the 12 by 3 rectangle's area of 36 square cm, so the triangles tile it perfectly.

Instead of guess and check, reason directly: the only single-row rectangle of height 3 cm with perimeter 30 cm must be 12 cm long, which is 4 squares, hence 8 triangles.

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing that two congruent right triangles form a square.
3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying squares by 2 triangles each and scaling row length.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Setting the rectangle's perimeter equal to 30 cm to find the length.

💡 Two matching triangles make one little square, so once you find the rectangle you just double the squares to count triangles!