Geometry

Key Takeaways

Memorize key formulas: Area of triangle = 1/2 base times height; Area of circle = pi times r^2.
The sum of angles in a triangle is always 180 degrees; in a quadrilateral, 360 degrees.
The Pythagorean theorem (a^2 + b^2 = c^2) applies only to right triangles.
Know the special right triangles: 30-60-90 (1, sqrt(3), 2) and 45-45-90 (1, 1, sqrt(2)).
For volume problems, multiply the base area by height for prisms and cylinders.

Last updated: January 2026

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

Angles

Types of Angles

Type	Degrees	Example
Acute	Less than 90°	45°
Right	Exactly 90°	90°
Obtuse	Between 90° and 180°	120°
Straight	Exactly 180°	180°

Angle Relationships

Relationship	Property
Complementary	Two angles that sum to 90°
Supplementary	Two angles that sum to 180°
Vertical	Opposite angles formed by intersecting lines (equal)

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

By Sides	Description	By Angles	Description
Equilateral	All sides equal	Acute	All angles < 90°
Isosceles	Two sides equal	Right	One angle = 90°
Scalene	No sides equal	Obtuse	One angle > 90°

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!)

Triple	Multiple Examples
3, 4, 5	6-8-10, 9-12-15, 12-16-20
5, 12, 13	10-24-26
8, 15, 17
7, 24, 25

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

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

Worked Example: Triangle Area

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

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

Quadrilaterals

Quadrilateral Properties

Shape	Properties	Area Formula
Square	4 equal sides, 4 right angles	$s^{2}$
Rectangle	Opposite sides equal, 4 right angles	$l \times w$
Parallelogram	Opposite sides parallel and equal	$b \times h$
Trapezoid	One pair of parallel sides	$\frac{1}{2} (b_{1} + b_{2}) \times h$

Worked Example: Trapezoid Area

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

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

Circles

Circle Formulas

Measurement	Formula
Circumference	$C = 2 π r = π d$
Area	$A = π r^{2}$

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

Key Terms

Radius (r): Distance from center to edge
Diameter (d): Distance across through center ( $d = 2 r$ )
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: $C = 2 π (7) = 14 π \approx 44$ units
Area: $A = π (7)^{2} = 49 π \approx 154$ square units

Volume and Surface Area

3D Shape Formulas

Shape	Volume	Surface Area
Cube	$s^{3}$	$6 s^{2}$
Rectangular Prism	$l \times w \times h$	$2 (l w + l h + w h)$
Cylinder	$π r^{2} h$	$2 π r^{2} + 2 π r h$
Sphere	$\frac{4}{3} π r^{3}$	$4 π r^{2}$
Cone	$\frac{1}{3} π r^{2} h$	$π r^{2} + π r s$

Worked Example: Cylinder Volume

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

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

Test Your Knowledge

A right triangle has legs of 5 and 12. What is the length of the hypotenuse?

Test Your Knowledge

What is the area of a circle with radius 5? (Use pi = 3.14)

15.7 sq units
31.4 sq units
78.5 sq units
157 sq units

Test Your Knowledge

Two angles are supplementary. If one angle measures 115 degrees, what is the measure of the other angle?

25 degrees
65 degrees
75 degrees
115 degrees

Test Your Knowledge

A rectangular box has dimensions 4 x 5 x 6. What is its volume?

15 cubic units
60 cubic units
120 cubic units
148 cubic units