ACT Math || SAT Math || 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?
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 (rotated square) whose area is
  $$ 2R^2 $$
• For \( a|x-h| + b|y-k| = c \), the diagonals are
  $$ d_x = \frac{2c}{a}, \qquad d_y = \frac{2c}{b} $$
  and area is
  $$ A = \frac{d_x d_y}{2} $$
• Distance from origin to line \( Ax+By+C=0 \) is
  $$ \frac{|C|}{\sqrt{A^2+B^2}} $$
• Area of an inscribed square in a circle of radius \(r\):
  $$ A = 2r^2 $$
• Area of a regular hexagon inscribed in a circle of radius \(r\):
  $$ A = \frac{3\sqrt{3}}{2}r^2 $$

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

Created for practice — absolute-value regions, circles, and inscribed figures.