Reasoning · Grade 3-2 Circles and Length

Problem

Perimeter of a figure enclosing circles

Identical 20 in plates fill a U-shaped table, whose edges sit flush against them. Four run along the bottom and three up each side. Every edge is a whole number of plate diameters. Find the table's perimeter.
Your answer
How to solve
Strategy Draw a Diagram — Sketch the U and label every edge in plate-diameters first (Draw a Diagram). The U is a big rectangle with a rectangular notch, so I find the big rectangle's size and the notch's size as separate subproblems (Identify Subproblems). Tracking one plate = 20 in lets me turn every counted diameter into inches at the end (Analyze the Units).
1STEP 1

Find the overall width and height in plates

In plates the table spans 4 across and 3 up.

width=4 plates, height=3 plates
2STEP 2

Find the notch (the open top middle) in plates

The U is that rectangle with a notch cut from the top middle.

notch width=4-2=2 plates, notch depth=3-1=2 plates
3STEP 3

Walk around the U and add the edges (in plates)

Walking the border counts 18 plate-widths.

4+3+1+2+2+2+1+3=18 plates
4STEP 4

Convert plate-diameters to inches

At 20 in each, the perimeter is 360 in.

18 × 20=360 in
Answer
360 in
18 × 20 = 360
Cross-check with the shortcut for a notched rectangle: a plain 4-by-3 plate rectangle has perimeter 2 x (4 + 3) = 14 plates; cutting a rectangular notch into one edge adds two notch-side walls of depth 2 each (the notch bottom replaces the removed top edge of equal length), so 14 + 2 + 2 = 18 plates = 360 in. Both methods agree, and 360 in is a sensible size for a table holding 8 dinner plates.
Takeaway

Count the table's edges in plate-widths first (18 of them), then multiply by 20 in once - perimeter is just one big trip around!

  • Find the overall width and height in plates
  • Find the notch (the open top middle) in plates
  • Walk around the U and add the edges (in plates)
  • Convert plate-diameters to inches