Problem

Perimeter plus tiles needed to build it

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.

Measurement & dataOperations

Your answer

triangles

How to solve

Strategy Draw a Diagram — 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.

1STEP 1

Combine two triangles into a square

Two triangles joined along the hypotenuse make a 3 cm by 3 cm square.

2 triangles = 1 square (3 cm × 3 cm)

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

3.G.A.1Draw A Diagram

Two copies of this right triangle, joined along their slanted sides, fit together into a 3 cm by 3 cm square.

Why?

The two triangles meet along their slanted sides with no gap and no overlap, so together they cover one whole four-sided piece.

Why?

The second triangle is an exact copy of the first, so its slanted side is the same length and lies exactly on the first one.

🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

Why?

Two pieces set together with no gap and no overlap combine into one whole shape.

🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

Why?

The four outside edges of the joined shape are the four legs of the triangles, and every leg is 3 cm, so all four sides are equal.

Why?

The second triangle is an exact copy of the first, so its legs are 3 cm just like the first triangle's legs.

🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

Why?

Every corner of the joined shape is a right angle, so a shape with four equal sides and four right angles is a square.

Why?

Two corners are the triangles' own right angles, and the other two corners are each made of two equal slant angles that together fill a right angle.

Why?

The three corners of any triangle add up to 180 degrees, so alongside the 90 degree corner the two slant angles add up to 90 degrees.

🧱A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.

Why?

The two legs are the same length, so folding the triangle down the middle lays one slant corner exactly on the other, making the two slant angles equal at 45 degrees each.

🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

2STEP 2

Line the squares up in a row

Join n of these squares in a row: the rectangle is 3 cm tall and 3 times n cm long.

length = 3 × n, height = 3

3STEP 3

Match the perimeter to 30 cm

Set the perimeter 2(3n + 3) = 30 and solve: 3n + 3 = 15, so n = 4, giving a 12 cm by 3 cm rectangle.

2 (3n + 3) = 30 → 3n + 3 = 15 → n = 4

4STEP 4

Count the triangles

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

4 × 2 = 8

Answer

8 triangles

4 × 2 = 8

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.

Takeaway

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

Combine two triangles into a square
Line the squares up in a row
Match the perimeter to 30 cm
Count the triangles

Where next?

Level upsuggested

Perimeter of overlapped squares and rectangles Distance between parallel sides equals the side length

Another one like thissuggested

9 practice problems

Same look, different logicsuggested

Count all shapes hidden in a grid Count distinct shapes built by adding congruent tiles

Step back and solidifysuggested

Find a side from equal perimeters Side lengths from overlapping rectangles

1 reasoning problems grow out of this type — the bridge toward competition →

▶ Practice — 9 problems