Geometry Formulas: Complete Reference for Students

Grades 9-12 geometry formulas split into two groups: flat 2D figures use perimeter and area, while 3D solids use surface area and volume. This reference lists each formula with its symbols, units, and a worked example. Start by matching the measurement the problem asks for, pick the shape, substitute the values, and check that the units match: plain units for length, square units for area, and cubic units for volume.

A Grades 9-12 geometry course returns to the same set of formulas whether the task is homework, a quiz, or a state test. The formulas fall into three families: measurements for flat shapes, measurements for solid objects, and the relationships that connect side lengths and coordinates. If you need the definitions behind these formulas first, the geometry concepts guide explains points, lines, angles, and the shapes themselves; this page keeps the formulas, their symbols, and the checks that keep the arithmetic correct.

Before choosing a formula, decide what the problem measures. Perimeter and circumference are lengths around a flat figure, so they carry plain units such as meters or centimeters. Area covers the inside of a flat figure and carries square units such as m² or cm². On a solid, surface area covers the outside faces and also uses square units, while volume fills the space inside and uses cubic units such as m³ or cm³. LibreTexts keeps the same split: length measurements for 2D outlines, square units for 2D area, and cubic units for 3D volume.

A problem that asks "how far around?" wants perimeter or circumference; "how much surface?" wants area or surface area; "how much space inside?" wants volume.

Flat shapes live on a plane and are measured by perimeter and area. The tables below list the shapes a standard Grades 9-12 course uses most often, with the symbol for each measurement.

For polygons, perimeter adds the side lengths, and area uses a base with a perpendicular height. The table lists the shapes a standard Grades 9-12 course uses most often.

The height in each area formula sits perpendicular to the base, not along the slanted side. For a triangle, that perpendicular height may fall outside the shape when the triangle is obtuse, but the formula stays the same.

A circle is measured by radius r, the distance from center to edge, and diameter d = 2r, the distance across through the center.

The π factor appears in both formulas, but area squares the radius, so substituting diameter for radius changes the result by a factor of four.

A right triangle carries one 90-degree angle, and the side opposite that angle is the hypotenuse c. The Pythagorean theorem relates the two legs a and b to the hypotenuse: a² + b² = c². The theorem applies only to right triangles; for any other triangle, use the area formula above and a separate law for side lengths that this reference does not expand.

Solids occupy space and are measured by surface area and volume. The NYSED Geometry Reference Sheet lists the same standard notation used on state Regents exams: general prisms and cylinders use V = Bh, cones and pyramids use V = ⅓Bh, and a sphere uses V = 4/3 πr³.

A cube with side s has six square faces, and a rectangular prism has length l, width w, and height h.

For a general prism, B is the area of the base and P is the perimeter of that base; the shape of the base does not change V = Bh.

A cone holds one third of the cylinder with the same radius and height. Its curved surface uses slant height ℓ rather than perpendicular height h. The 4/3 coefficient in sphere volume is easy to drop, so write it before substituting numbers.

A pyramid with base area B and perpendicular height h has volume V = ⅓Bh, one third of the prism that shares the same base and height. The base can be a square, triangle, or any polygon; only B changes, while the ⅓ factor stays the same.

On an x-y grid, the Pythagorean theorem becomes the distance formula. For two points (x₁, y₁) and (x₂, y₂), the distance between them is d = √((x₂ − x₁)² + (y₂ − y₁)²). The midpoint, the point halfway between them, is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Check with points (1, 2) and (4, 6): d = √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5, and the midpoint is (2.5, 4).

Each example below solves on its own, then shows a check.

Example 1 — Rectangle perimeter and area. A rectangle measures 12 m by 8 m. Perimeter P = 2(12 + 8) = 40 m. Area A = 12 × 8 = 96 m². Perimeter carries meters, area carries square meters, and 12 × 8 matches the two dimensions.

Example 2 — Circle circumference and area. A circle has radius 5 cm. Circumference C = 2π(5) = 10π ≈ 31.42 cm. Area A = π(5)² = 25π ≈ 78.54 cm². Both formulas use r = 5, not d = 10, and area squares the radius.

Example 3 — Triangle area. A triangle has base 10 in and perpendicular height 6 in. Area A = ½(10)(6) = 30 in². Keep the ½ factor, and make sure the height is perpendicular to the base.

Example 4 — Cylinder volume and surface area. A cylinder has radius 3 and height 10. Volume V = π(3)²(10) = 90π ≈ 282.74 cubic units. Surface area SA = 2π(3)² + 2π(3)(10) = 18π + 60π = 78π ≈ 245.04 square units. Volume uses cubic units, surface area uses square units, and the two ends appear once each.

Example 5 — Sphere volume and surface area. A sphere has radius 3. Volume V = 4/3 π(3)³ = 36π ≈ 113.10 cubic units. Surface area SA = 4π(3)² = 36π ≈ 113.10 square units. The 4/3 coefficient stays in volume, and surface area uses 4πr².

Example 6 — Pythagorean theorem. A right triangle has legs 6 and 8. c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10. The triangle has a right angle, and 10 is longer than both legs.

Using diameter instead of radius tops the list: d = 2r, so divide before substituting into πr² or 2πr. Units drift next: perimeter and circumference stay in plain units, area and surface area stay in square units, and volume stays in cubic units. On a cone or pyramid, slant height ℓ and perpendicular height h are different measurements; volume uses h, while curved-surface area uses ℓ. Sphere volume loses the 4/3 coefficient, and cone or pyramid volume loses the ⅓ factor, when students copy the formula too fast. The Pythagorean theorem does not apply to a triangle without a right angle. After each calculation, write the unit, restate the formula you used, and recheck the arithmetic on paper. Khan Academy offers free practice units on area, volume, the Pythagorean theorem, and circles when you want a second set of problems.

The existing geometry course follows units on area and perimeter, volume and surface area, the Pythagorean theorem, circles, and coordinate geometry, so you can match a unit to the formula you are practicing. If the available units do not match your syllabus, open Gauth Math Courses, select Create Course, enter your topic and level, and generate a focused interactive micro-course. Return to a worked example after a lesson and solve it again on your own.