Geometry & Figures

Problem

All radii in one circle are equal

A square is drawn inside a circle so that all four corners sit on the circle. The square's perimeter is 40 cm, and the radius from the circle's center to a corner is 7 cm. The diagonal of the square passes through the center, so it is a diameter. I need the perimeter of the triangle formed by one side of the square, an adjacent side, and that diagonal.
7 cm A B C D
Measurement & dataGeometry
Your answer
cm
How to solve
Strategy Draw a Diagram — The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. I find each side as a small subproblem, then add the three sides.
1STEP 1

Find one side of the square

A square has four equal sides, so each side is the perimeter divided by 4.

40 ÷ 4 = 10
2STEP 2

Find the diagonal (a diameter)

The diagonal ㄹㄴ goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 7 cm.

7 × 2 = 14
3STEP 3

Add the three sides of the triangle

Triangle ㄹㄴㄷ has two sides that are sides of the square (ㄹㄷ and ㄴㄷ, each 10 cm) plus the diagonal ㄹㄴ (14 cm). Add them for the perimeter.

10 + 10 + 14 = 34
Answer
34 cm
10 + 10 + 14 = 34
The diagonal (14 cm) is the longest side, which matches the picture where the diameter stretches across the square. Two sides of 10 plus a 14 give 34 cm, a sensible perimeter that is larger than the square's side and smaller than the square's full perimeter.
Takeaway

This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

  • Find one side of the square
  • Find the diagonal (a diameter)
  • Add the three sides of the triangle
Where next?
Another one like thissuggested
10 practice problems
Step back and solidifysuggested
Center to edge distance is the radiusPerimeter splits into radius arcs and straight parts
3 reasoning problems grow out of this type — the bridge toward competition →

▶ Practice — 10 problems