Use equilateral 60-degree angles and equal sides

4.G.A.24.MD.C.7 · take · grade 4

Archetype: Isosceles and Equilateral Angle Chaining · step in a 6-type progression

You want to make an equilateral triangle whose three side lengths add up to the same total as the isosceles triangle below. How many centimeters should each side of the equilateral triangle be?

Show solution

Understand

An isosceles triangle has two equal slant sides of 21 cm and a base of 15 cm. I need to build an equilateral triangle whose total side length (perimeter) is the same, and find how long each of its equal sides must be.

Givens

The isosceles triangle has two equal sides of 21 cm each
Its remaining (base) side is 15 cm
The equilateral triangle must have the same perimeter as the isosceles triangle

Unknowns

The length of one side of the equilateral triangle (in cm)

Constraints

An equilateral triangle has all three sides equal
Total perimeters of the two triangles must be equal

Plan

#7 Identify Subproblems · also uses: #8 Analyze the Units

Split the task into two small steps: first find the isosceles triangle's perimeter by adding its three sides, then share that total equally among the equilateral triangle's three sides. Checking centimeters throughout keeps the setup correct.

Execute

#7 Identify Subproblems 4.MD.A.3

An isosceles triangle has two equal sides, so both slant sides are 21 cm. Add all three sides to get the perimeter.

21 + 21 + 15 = 57\ \text{cm}

Grade 4 students find a perimeter by adding the side lengths around the shape.

#7 Identify Subproblems 4.G.A.2

The equilateral triangle must have the same 57 cm perimeter, and all three of its sides are equal, so divide the perimeter by 3.

57 \div 3 = 19\ \text{cm}

Equal sides means simply splitting the total length into 3 equal parts, a Grade 4 division idea.

#8 Analyze the Units 4.MD.A.1

Each side of the equilateral triangle is 19 cm. Units stay in centimeters from start to finish.

\text{each side} = 19\ \text{cm}

Keeping the unit (cm) consistent confirms the answer is a length, not a count.

Answer: 19 cm

Review

Check: 19 + 19 + 19 = 57 cm, which matches the isosceles perimeter of 21 + 21 + 15 = 57 cm. Each equilateral side (19 cm) lands neatly between the isosceles base (15 cm) and slant side (21 cm), which makes sense.

Guess and check (tool 6): try 19 cm per side, multiply by 3 to get 57 cm, and confirm it equals the isosceles perimeter; adjust the guess up or down if it didn't match.

Standards · min grade 4

4.MD.A.3 Apply area and perimeter formulas for rectangles in real world and mathematical problems — Finding the perimeter of the isosceles triangle by adding its side lengths.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the equilateral triangle's property that all three sides are equal.
4.MD.A.1 Know relative sizes of measurement units and convert within a system — Keeping the length unit (centimeters) consistent throughout.

💡 This only needs Grade 4 perimeter-adding and equal-sharing you already know!