SHSAT Geometry prep

The geometry topics that matter

Lines and angles

Basic angle facts:

• A straight line is 180°. Angles on a straight line sum to 180°.
• Vertical angles (formed by intersecting lines) are equal.
• When parallel lines are cut by a transversal: corresponding angles are equal, alternate interior angles are equal, co-interior angles sum to 180°.
• Angles around a point sum to 360°.

SHSAT questions on lines and angles often combine these facts in 2–3 step problems. The skill is recognizing which fact applies in a given diagram.

Triangles

Triangle facts that appear constantly:

• The three angles of a triangle sum to 180°.
• The longest side is opposite the largest angle; the shortest side is opposite the smallest angle.
• An isosceles triangle has two equal sides and two equal angles (opposite the equal sides).
• An equilateral triangle has three equal sides and three 60° angles.
• The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

The Pythagorean theorem

For a right triangle with legs a and b and hypotenuse c: a² + b² = c². This is one of the most frequently used facts in SHSAT geometry. Common Pythagorean triples worth memorizing:

• 3-4-5 (and multiples: 6-8-10, 9-12-15, etc.)
• 5-12-13
• 8-15-17
• 7-24-25

Recognizing a triple lets you solve right-triangle problems by inspection rather than computation, saving substantial time on the test.

Special right triangles

Two right triangles with memorable side ratios:

• 45-45-90: Side ratios are 1 : 1 : √2. The legs are equal; the hypotenuse is leg × √2.
• 30-60-90: Side ratios are 1 : √3 : 2. The short leg (opposite 30°) is half the hypotenuse; the long leg (opposite 60°) is short leg × √3.

Quadrilaterals

The four-sided shapes and their properties:

• Square: 4 equal sides, 4 right angles. Area = s².
• Rectangle: Opposite sides equal, 4 right angles. Area = l × w.
• Parallelogram: Opposite sides parallel and equal. Opposite angles equal. Area = base × height.
• Rhombus: 4 equal sides. Opposite angles equal. Diagonals perpendicular and bisecting.
• Trapezoid: One pair of parallel sides. Area = (1/2)(b₁ + b₂)h.

Circles

Circle facts:

• Circumference = 2πr = πd
• Area = πr²
• A sector with central angle θ (in degrees) has area = (θ/360) × πr² and arc length = (θ/360) × 2πr
• Inscribed angles equal half the central angle subtending the same arc

Volume and surface area

3D shape formulas worth knowing:

• Rectangular prism (box): Volume = lwh; Surface area = 2(lw + lh + wh)
• Cube: Volume = s³; Surface area = 6s²
• Cylinder: Volume = πr²h; Surface area = 2πr² + 2πrh

Cones and spheres appear less often but are worth knowing if you have time.

Coordinate geometry basics

The SHSAT tests basic coordinate geometry:

• Distance between two points: √((x₂−x₁)² + (y₂−y₁)²) — which is just the Pythagorean theorem applied to coordinates
• Midpoint of a segment: ((x₁+x₂)/2, (y₁+y₂)/2)
• Slope of a line: (y₂−y₁)/(x₂−x₁)
• Slope-intercept form: y = mx + b

How to study geometry for the SHSAT

• Memorize the formulas. Unlike algebra, where techniques can sometimes be reasoned out, geometry depends heavily on knowing specific formulas. Spend time memorizing the basic ones until they're automatic.
• Practice with diagrams. Most geometry questions include diagrams. Get used to extracting information from them and adding your own annotations.
• Recognize Pythagorean triples. Saving 30 seconds on each of 4–5 right-triangle questions adds up to meaningful time savings on the section.
• Don't panic on combination problems. Many harder geometry questions combine two or three concepts (e.g., a triangle inside a circle, area of a sector minus area of a triangle). Take them step by step.