Competitive exam || Area on the Coordinate Plane || Practice Set

Area on the Coordinate Plane

Practice Set — Absolute Values, Circles & Inscribed Shapes

Instructions

• You may use scratch paper and a calculator where appropriate.
• Show all steps — for absolute-value regions, identify vertices or diagonals; for circles give radius/diameter; for inscribed shapes justify your geometry.
• Answers requiring \(\pi\) may be left in terms of \(\pi\).

Problems (1–15)

1. What is the area, in square units, enclosed by \[ y = -|x+4| + 7 \quad\text{and}\quad y = |x+4| - 5 ? \]
2. What is the area of the region enclosed by \[ y = 12 - |x-6|. \] (Interpret the V-shape and the x-intercepts.)
3. What is the area of the region described by \[ |x-8| + |y-1| = 10\; ? \]
4. Find the area of the region enclosed by \[ 3|x| + 2|y| = 18. \]
5. What is the absolute difference between the lengths of the diagonals of the region \[ 4|x-2| + |y+3| = 20\; ? \]
6. What is the area, in square units, enclosed by \[ (x-4)^2 + (y-9)^2 = 25 ? \] Express your answer in terms of \(\pi\).
7. The area enclosed by \[ |x-a| + |y-2| = b \] is \(72\) square units. Find \(b\).
8. If the region enclosed by \[ 6|x+1| + 3|y-5| = k \] has area \(54\), find the value of \(k\).
9. A circle is inscribed in the figure \[ 7|x| + 24|y| = 84. \] What is the area of the circle? Express your answer as a common fraction in terms of \(\pi\).
10. A square is inscribed in the circle \[ (x-3)^2 + (y+4)^2 = 32. \] What is the area of the square?
11. What is the area of the region enclosed by \[ y = |x+10| - 3 \quad\text{and}\quad y = -|x+10| + 9 ? \]
12. What is the area, in square units, of the region enclosed by \[ |x-2| + 3|y| = 15\; ? \]
13. A diamond described by \[ 2|x-1| + 5|y+6| = p \] has diagonals whose lengths differ by \(6\). Find \(p\).
14. The graph \[ (x+12)^2 + (y-1)^2 = 18 \] is a circle. What is the area of a regular hexagon inscribed in this circle? (Leave your answer in terms of \(\pi\).)
15. A rectangle is inscribed in the region \[ |x| + |y| = 14, \] with its sides parallel to the coordinate axes. What is the maximum possible area of such a rectangle? Explain your reasoning.

Hints & Helpful Formulas

• For \(|x-h| + |y-k| = R\) the graph is a diamond (a rotated square) whose area is \(2R^2\).
• For \(a|x-h| + b|y-k| = c\), the diamond's horizontal half-width is \(c/a\) and vertical half-height is \(c/b\). Full diagonals: \(d_x = 2c/a,\ d_y = 2c/b.\) Area \(= \dfrac{d_x d_y}{2}.\)
• Distance from origin to line \(Ax+By+C=0\) is \(\dfrac{|C|}{\sqrt{A^2+B^2}}\) when the line is written appropriately; use this to compute an inscribed circle's radius.
• Area of an inscribed square in a circle of radius \(r\): diagonal \(=2r\) so area \(= \dfrac{(2r)^2}{2} = 2r^2.\)
• Area of a regular hexagon inscribed in a circle of radius \(r\): \( \dfrac{3\sqrt{3}}{2}r^2\). (Useful for problem 14.)

Use the hints if you get stuck, but show the full steps in your written work.

Created for practice — absolute-value regions, circles, and inscribed figures. If you'd like, I can also generate a printable PDF, provide a worked answer key, or produce step-by-step solutions for each problem.