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Plane geometry (polygons, triangles, angles) — ISEE Upper practice questions

On the ISEE Upper Mathematics Achievement section, Plane geometry (polygons, triangles, angles) covers figures in the plane such as circles, squares, rectangles, and right triangles, along with the angle and side relationships needed to find lengths and areas. A student must convert between circumference and area, find a rectangle's perimeter from given vertices, and compute how much of a square remains after congruent right triangles are removed from its corners. Items typically ask for a single numerical result. Answer choices often include values that mix radius with diameter, confuse remaining area with the part that was cut away, or use an incomplete set of sides for perimeter.

  • Section: Mathematics Achievement 47 questions on the paper
  • 134 practice questions in the app

Sample questions

Question 1

A circle has circumference 18π18\pi. What is the area of the circle?

  1. A81π81\pi
  2. B36π36\pi
  3. C162π162\pi
  4. D324π324\pi
Show answer

Answer: A — 81π81\pi

Since C=2πrC=2\pi r, 18π=2πr18\pi=2\pi r, so r=9r=9. The area is πr2=π(92)=81π\pi r^2=\pi(9^2)=81\pi.

Question 2

A square has side length 12 units. Four congruent right triangles are removed from the square, one from each corner. Each removed triangle has legs of length 3 units and 4 units along the sides of the square. What percent of the square's original area remains?

  1. A8313%83\dfrac{1}{3}\%
  2. B80%80\%
  3. C8712%87\dfrac{1}{2}\%
  4. D9123%91\dfrac{2}{3}\%
Show answer

Answer: A — 8313%83\dfrac{1}{3}\%

The square's area is 12⋅12=14412\cdot12=144. Each removed triangle has area 12⋅3⋅4=6\dfrac{1}{2}\cdot3\cdot4=6, so the four triangles have total area 24. The remaining area is 144−24=120144-24=120, and 120144=56=8313%\dfrac{120}{144}=\dfrac{5}{6}=83\dfrac{1}{3}\%.

Question 3

Three vertices of an axis-aligned rectangle are (−5,−3)(-5,-3), (1,−3)(1,-3), and (1,1)(1,1). What is the rectangle's perimeter?

  1. A10
  2. B20
  3. C22
  4. D24
Show answer

Answer: B — 20

The horizontal and vertical side lengths are 6 and 4. Therefore the perimeter is 2(6+4)=202(6+4)=20.

Practice 134 Plane geometry (polygons, triangles, angles) questions in the app

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