Domain 3 • Spatial & Analytical Geometry
Geometry on ALEKS
Geometry questions on the ALEKS placement exam test both classic Euclidean concepts (triangles, angles, solids) and analytic coordinate geometry (distance, midpoint, circle equations).
Core Tested Concepts
Right Triangles & Pythagorean Theorema² + b² = c², recognizing common Pythagorean triples (3-4-5, 5-12-13, 8-15-17), special right triangles (45-45-90 and 30-60-90).
2D Perimeter & Area FormulasArea of triangles (1/2 bh), trapezoids (1/2(b1+b2)h), circles (πr²), and circumference (2πr).
3D Solids: Volume & Surface AreaRectangular prisms (l·w·h), cylinders (πr²h), cones (1/3 πr²h), and spheres (4/3 πr³).
Coordinate Geometry & CirclesDistance formula, midpoint formula ((x1+x2)/2, (y1+y2)/2), and the standard equation of a circle: (x - h)² + (y - k)² = r².
High-Yield Cheat Sheet
Essential Geometry Formulas
Distance Formula
d = √((Δx)² + (Δy)²)
Circle Equation
(x-h)² + (y-k)² = r²
Cylinder Volume
V = πr²h
Practice Drills
5 Authentic Geometry Practice Questions
1Practice Problem
A right triangle has a hypotenuse of length 13 cm and one leg of length 5 cm. What is the length of the other leg?
Hint: Use the Pythagorean theorem: a² + b² = c². Solve for a: a² = 13² - 5².
Formula: a² + b² = c²
2Practice Problem
What is the area of a circle whose diameter is 10 units? (Give your answer in terms of π)
Hint: The radius is half the diameter: r = 10 / 2 = 5. Area = πr².
Formula: A = πr², r = d/2
3Practice Problem
Calculate the distance between the points (1, 2) and (4, 6) in the Cartesian coordinate plane.
Hint: Distance d = √((x2 - x1)² + (y2 - y1)²).
Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
4Practice Problem
Find the volume of a rectangular box with length 8 cm, width 5 cm, and height 4 cm.
Hint: Volume of a rectangular prism = length × width × height.
Formula: V = l · w · h
5Practice Problem
What are the coordinates of the center (h, k) and radius r of the circle given by (x - 3)² + (y + 5)² = 49?
Hint: Standard circle equation is (x - h)² + (y - k)² = r². Notice y + 5 = y - (-5).
Formula: (x - h)² + (y - k)² = r²