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Mathematics Achievement

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

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What this topic is

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.

Sample questions

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Question 1Easier

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

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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 2Mid

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?

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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 3Mid

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?

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Answer: B — 20

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

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Practice 136 Plane geometry (polygons, triangles, angles) questions in the app

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