Geometry

MK Subtest: Geometry questions make up about 30-35% of the Mathematics Knowledge section. Memorize the key formulas!

Angles

Types of Angles

Angle Relationships

Worked Example: Complementary Angles

Problem: Two angles are complementary. One angle is 35°. What is the other?

Solution: $90° - 35° = 55°$

Triangles

Triangle Angle Sum

The sum of all angles in a triangle is always 180°.

Types of Triangles

The Pythagorean Theorem

For right triangles only:

$a^2 + b^2 = c^2$

where $c$ is the hypotenuse (longest side, opposite the right angle)

Common Pythagorean Triples (Memorize These!)

Worked Example: Pythagorean Theorem

Problem: A right triangle has legs of 6 and 8. Find the hypotenuse.

Solution: Recognize 6-8-? is a multiple of 3-4-5 (multiplied by 2):

• If 3-4-5, then 6-8-10

Or calculate: $c^2 = 6^2 + 8^2 = 36 + 64 = 100$, so $c = 10$

Special Right Triangles

45-45-90 Triangle (Isosceles Right):

• Sides in ratio: $1 : 1 : \sqrt{2}$
• If legs = 5, hypotenuse = $5\sqrt{2}$

30-60-90 Triangle:

• Sides in ratio: $1 : \sqrt{3} : 2$
• Opposite 30°: shortest side (1)
• Opposite 60°: middle side ($\sqrt{3}$)
• Opposite 90°: longest side (2)

Triangle Area

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Worked Example: Triangle Area

Problem: Find the area of a triangle with base 12 and height 8.

Solution: $\text{Area} = \frac{1}{2} \times 12 \times 8 = 48$ square units

Quadrilaterals

Quadrilateral Properties

Worked Example: Trapezoid Area

Problem: A trapezoid has parallel sides of 8 and 12, with height 5. Find the area.

Solution: $\text{Area} = \frac{1}{2}(8 + 12) \times 5 = \frac{1}{2}(20) \times 5 = 50$ square units

Circles

Circle Formulas

Approximation: Use $\pi \approx 3.14$ or $\frac{22}{7}$ for calculations

Key Terms

• Radius (r): Distance from center to edge
• Diameter (d): Distance across through center ($d = 2r$)
• Chord: Line segment connecting two points on the circle
• Arc: Portion of the circumference

Worked Example: Circle Calculations

Problem: A circle has radius 7. Find its circumference and area.

Solution:

• Circumference: $C = 2\pi(7) = 14\pi \approx 44$ units
• Area: $A = \pi(7)^2 = 49\pi \approx 154$ square units

Volume and Surface Area

3D Shape Formulas

Worked Example: Cylinder Volume

Problem: Find the volume of a cylinder with radius 3 and height 10.

Solution: $V = \pi r^2 h = \pi (3)^2 (10) = 90\pi \approx 283$ cubic units