Perimeter of overlapped squares and rectangles

4.MD.A.34.G.A.2 · adapt · grade 4

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

The figure below is made by joining a rectangle and a square edge to edge with no overlap. Find the sum of the lengths of the four sides (the perimeter) of rectangle ABCD, in cm.

Figure description: A wide rectangle ABCD (A top-left, D top-right, B bottom-left, C bottom-right) is divided into two parts by a vertical segment MN joining a point M on the top edge to a point N on the bottom edge. The left part AMNB is a rectangle whose top side AM measures $4\text{ cm}$ , and the right part MDCN is a square. The left side AB is marked $10\text{ cm}$ .

(Note: metric units are kept because the answer depends on the figure's cm labels, which are not being redrawn for this localization.)

Show solution

Understand

A big rectangle ABCD is split by a vertical segment MN into a left rectangle AMNB and a right square MDCN. The top side AM of the left rectangle is 4 cm and the left side AB is 10 cm. I need the perimeter of the big rectangle ABCD.

Givens

ABCD is a rectangle with A top-left, D top-right, B bottom-left, C bottom-right.
A vertical segment MN splits ABCD into left part AMNB and right part MDCN.
The left part AMNB is a rectangle; the right part MDCN is a square.
Top side AM = 4 cm.
Left side AB = 10 cm.

Unknowns

The perimeter of rectangle ABCD, in cm.

Constraints

A square has four equal sides.
In a rectangle, opposite sides are equal.
The height of the whole figure equals AB = MN = 10 cm.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Use the square to find the missing length. The dividing segment MN equals the height AB = 10 cm, and because MDCN is a square, every side of it is 10 cm, so MD = 10 cm. Then the full top AD = AM + MD, and the rectangle's perimeter is twice the length plus twice the width.

Execute

#1 Draw a Diagram 4.G.A.1

The left side AB is 10 cm, and the dividing segment MN is the same height as the rectangle, so MN = 10 cm.

MN = AB = 10 \text{ cm}

MN runs from the top edge to the bottom edge, so it is exactly as tall as the rectangle.

#7 Identify Subproblems 4.G.A.2

The right part MDCN is a square, so all its sides are equal. Its side MN is 10 cm, so MD = 10 cm.

MD = MN = 10 \text{ cm}

A square's four sides are all the same length, so once one side is 10 cm they all are.

#7 Identify Subproblems 4.MD.A.3

The top of the big rectangle is AM plus MD.

AD = AM + MD = 4 + 10 = 14 \text{ cm}

The whole top edge is just the two top pieces laid end to end.

#7 Identify Subproblems 4.MD.A.3

The rectangle is 14 cm wide and 10 cm tall, so the perimeter is two widths plus two heights.

2 \times (14 + 10) = 2 \times 24 = 48 \text{ cm}

Going all the way around a rectangle covers each of the two lengths and two widths exactly once.

Answer: 48 cm

Review

The rectangle is 14 cm by 10 cm; a perimeter near 48 cm is right since 4 sides averaging 12 cm give about 48 cm. The width 14 cm is sensibly a bit more than the 10 cm height, matching a 'wide' rectangle.

Use Draw a Diagram (tool 1): walk the boundary adding 14 + 10 + 14 + 10 one side at a time to confirm 48 cm.

Standards · min grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying that MN spans the full height and equals AB.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the square's equal sides to get MD = MN = 10 cm.
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems — Computing AD = 14 cm and the perimeter 2(14+10) = 48 cm.

💡 Spot the square to fill in the missing length, then go around the rectangle: it only needs Grade 4 perimeter sense!