AMC 8 · 2005 · #4

Grade 5 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

perimeterarea-rectanglesmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: perimeterarea-rectangles

📏 Short solution 💡 2 insights

Problem

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: A square and a triangle have equal perimeters. The triangle's three sides are $6.1$ cm, $8.2$ cm, and $9.7$ cm. Find the area of the square in square centimeters.

Givens: Triangle side lengths: $6.1$ cm, $8.2$ cm, $9.7$ cm; The square and triangle have equal perimeters; Answer choices: (A) $24$, (B) $25$, (C) $36$, (D) $48$, (E) $64$

Unknowns: The area of the square in cm$^2$

Understand

Restated: A square and a triangle have equal perimeters. The triangle's three sides are $6.1$ cm, $8.2$ cm, and $9.7$ cm. Find the area of the square in square centimeters.

Givens: Triangle side lengths: $6.1$ cm, $8.2$ cm, $9.7$ cm; The square and triangle have equal perimeters; Answer choices: (A) $24$, (B) $25$, (C) $36$, (D) $48$, (E) $64$

Plan

Primary tool: #7 Break into Subproblems

Secondary: #1 Draw a Diagram

The question chains three small tasks behind one short sentence, so Tool #7 (Break into Subproblems) keeps the work organized: (i) add the three triangle sides to get the shared perimeter, (ii) divide by $4$ to get the square's side length, (iii) square that side to get the area. Tool #1 (Draw a Diagram) is the supporting move — a quick sketch of the triangle and square reminds us that "equal perimeter" is the only link between the two shapes, and that the square's four equal sides give the $\div 4$ step.

Execute — Answer: C

#7 Break into Subproblems 5.NBT.B.7 Step 1

Subproblem 1: find the triangle's perimeter by adding its three sides.

$$6.1 + 8.2 + 9.7 = 24 \text{ cm}$$

💡 Adding decimals to the tenths place is Grade 5 arithmetic — line up the decimal points and add column by column.

#7 Break into Subproblems 3.MD.D.8 Step 2

Subproblem 2: the square has the same perimeter, $24$ cm.
Since all four sides of a square are equal, divide the perimeter by $4$ to get one side.

$$\text{side} = \dfrac{24}{4} = 6 \text{ cm}$$

💡 The Grade 3 perimeter formula for a square is $P = 4s$, so $s = P/4$. Equal perimeters means the $24$ carries straight from the triangle to the square.

#7 Break into Subproblems 3.MD.C.7 Step 3

Subproblem 3: square the side to get the area.

$$\text{area} = 6 \times 6 = 36 \text{ cm}^2 \;\Rightarrow\; \textbf{(C)}$$

💡 Area of a square is side $\times$ side, a Grade 3 standard. A $6$-by-$6$ square holds $36$ unit squares.

[1] #7 5.NBT.B.7 Subproblem 1: find the triangle's perimeter by adding its three sides.

[2] #7 3.MD.D.8 Subproblem 2: the square has the same perimeter, $24$ cm. Since all four sides o

[3] #7 3.MD.C.7 Subproblem 3: square the side to get the area.

Review

Reasonableness: Cross-check the perimeter: $6.1 + 8.2 = 14.3$ and $14.3 + 9.7 = 24$, confirming $24$ cm. Then $24 \div 4 = 6$ and $6^2 = 36$, so (C) is consistent. The other choices fail a quick sanity test: (A) $24$ matches the perimeter, not the area; (E) $64 = 8^2$ would need a side of $8$, giving perimeter $32 \ne 24$; (D) $48$ isn't even a perfect square, so it can't be the area of a whole-number-sided square.

Alternative: Tool #1 (Draw a Diagram): sketch the triangle with sides labeled $6.1$, $8.2$, $9.7$ and walk around it counting cm — you arrive back at the start after $24$ cm. Draw a square next to it and walk around: $4$ equal sides totaling $24$ cm means each side is $6$ cm. The square visibly tiles into a $6 \times 6$ grid of unit squares, $36$ in all. Same answer (C).

CCSS standards used (min grade 5)

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Adding the three triangle sides $6.1 + 8.2 + 9.7 = 24$ with decimals aligned to the tenths place.)
3.MD.D.8 Solve real-world and mathematical problems involving perimeters of polygons (Using $P = 4s$ for the square to recover side $= 24/4 = 6$ cm from the shared perimeter.)
3.MD.C.7 Relate area to the operations of multiplication and addition (Computing the area of the $6$-cm square as $6 \times 6 = 36$ cm$^2$.)

⭐ When two shapes share a perimeter, that one number is the only bridge between them — add the triangle's sides to cross the bridge, then the square's $\div 4$ and side-squared are routine Grade 3 work.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 5)

More like this