Square
- Area
- A = s²
- Perimeter
- P = 4s
s side
Geometry Formulas
Every formula a geometry course uses, on one sheet: area and perimeter, circles, triangles, volume and surface area, and coordinates, each with a figure and every symbol named.
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Every formula a geometry course uses, on one page. Each shape has its figure, so you can see which length is the base and which is the height, and every symbol is named under the formula. Switch to quiz mode to hide the formulas and test yourself, or print the whole sheet.
Area and perimeter
Rectangle
- Area
- A = lw
- Perimeter
- P = 2l + 2w
l lengthw width
Triangle
- Area
- A = ½bh
- Perimeter
- P = a + b + c
b baseh heighta, b, c sides
Parallelogram
- Area
- A = bh
- Perimeter
- P = 2a + 2b
b baseh heighta side
Trapezoid
- Area
- A = ½(a + b)h
a, b parallel sidesh height
Rhombus
- Area
- A = ½d₁d₂
- Perimeter
- P = 4s
d₁, d₂ diagonalss side
Regular polygon
- Area
- A = ½aP
- Sum of interior angles
- S = (n − 2) · 180°
- Each interior angle
- (n − 2) · 180° / n
a apothemP perimetern number of sides
Circles
Circle
- Circumference
- C = 2πr = πd
- Area
- A = πr²
- d = 2r
r radiusd diameter
Arc and sector
- Arc length
- L = (θ / 360°) · 2πr
- Sector area
- A = (θ / 360°) · πr²
θ central angler radius
Equation of a circle
- (x − h)² + (y − k)² = r²
(h, k) centerr radius
Triangles
Pythagorean theorem
- a² + b² = c²
a, b legsc hypotenuse
Angles
- Angles of a triangle
- α + β + γ = 180°
- Complementary angles
- α + β = 90°
- Supplementary angles
- α + β = 180°
Special right triangles
- 30-60-90 sides
- x : x√3 : 2x
- 45-45-90 sides
- x : x : x√2
Equilateral triangle
- Area
- A = (√3 / 4)s²
- Height
- h = (√3 / 2)s
s side
Heron's formula
- Area
- A = √(s(s − a)(s − b)(s − c))
- s = (a + b + c) / 2
a, b, c sidess half the perimeter
Volume and surface area
Cube
- Volume
- V = s³
- Surface area
- SA = 6s²
s side
Rectangular prism
- Volume
- V = lwh
- Surface area
- SA = 2(lw + lh + wh)
l lengthw widthh height
Prism
- Volume
- V = Bh
- Surface area
- SA = 2B + Ph
B area of the baseP perimeter of the baseh height
Cylinder
- Volume
- V = πr²h
- Surface area
- SA = 2πr² + 2πrh
r radiush height
Cone
- Volume
- V = ⅓πr²h
- Surface area
- SA = πr² + πrℓ
- ℓ = √(r² + h²)
r radiush heightℓ slant height
Pyramid
- Volume
- V = ⅓Bh
- Surface area
- SA = B + ½Pℓ
B area of the baseP perimeter of the baseh heightℓ slant height
Sphere
- Volume
- V = ⁴⁄₃πr³
- Surface area
- SA = 4πr²
r radius
Coordinate geometry
Distance formula
- d = √((x₂ − x₁)² + (y₂ − y₁)²)
(x₁, y₁), (x₂, y₂) the two points
Midpoint formula
- M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
(x₁, y₁), (x₂, y₂) the two points
Slope
- m = (y₂ − y₁) / (x₂ − x₁)
- Parallel lines
- m₁ = m₂
- Perpendicular lines
- m₁ · m₂ = −1
m slope
Equation of a line
- Slope-intercept form
- y = mx + b
- Point-slope form
- y − y₁ = m(x − x₁)
m slopeb y-intercept
How to read the sheet
Find the shape, then match each letter in the formula to the figure. The letters are the usual ones in textbooks: b for a base, h for a height, r for a radius. A height is always measured at a right angle to the base, which is why the figures draw it dashed with a small square: on a slanted shape like a parallelogram, the height is not the slanted side.
Capital B means the area of a base, not a length. In V = Bh you first find the area of the end face (a triangle, a circle, any shape) and then multiply by the height.
Area, perimeter and volume answer different questions
Take a rectangle 3 cm by 4 cm. Its perimeter is the distance around it: 3 + 4 + 3 + 4 = 14 cm. Its area is how many 1 cm squares fit inside: 3 · 4 = 12 cm². Stand it up as a box 5 cm tall and the volume is how many 1 cm cubes fit inside: 3 · 4 · 5 = 60 cm³.
The units tell you which formula you need. A length times a length is an area; a length times a length times a length is a volume. If an answer about a volume comes out in square units, a factor is missing.
There are fewer formulas than it looks
Most of the sheet is a handful of ideas applied to different shapes.
- A triangle is half a parallelogram. Two copies of any triangle fit together into a parallelogram with the same base and height, so the area is ½bh.
- A trapezoid is a rectangle with the average width. Its area ½(a + b)h is the average of the two parallel sides times the height.
- Anything with the same cross section all the way up is base times height. A cube, a box, a prism and a cylinder all use V = Bh; only the base area changes.
- Anything that comes to a point is a third. A cone and a pyramid hold a third of the cylinder or prism around them, so V = ⅓Bh.
- Circles are all about π. The circumference is 2πr, the area is πr², and an arc or a sector is the same fraction of those as its angle is of 360°.
A worked example
A can is 10 cm tall with a radius of 3 cm. How much does it hold?
It is a cylinder, so V = πr²h. The base is a circle of area π · 3² = 9π, and the height is 10, so the volume is 90π, about 283 cm³. That is 283 ml: a little less than a standard soda can.
Try one
Where to go next
The formulas tell you what to compute, and the Pythagorean theorem calculator and the volume calculator will do the arithmetic and show each step. For the angles and ratios inside triangles, the trigonometry formula sheet is the next page.
Common questions
Is there a printable geometry formula sheet?
Yes. The sheet on this page is also a free letter-size PDF and a PNG, with the same figures and formulas. The download buttons are above the sheet and again under it.
What is the difference between area and surface area?
Area measures a flat shape: how much paper a rectangle covers. Surface area measures the outside of a solid: how much paper it takes to wrap a box. A box's surface area is the sum of the areas of its six faces, which is where SA = 2(lw + lh + wh) comes from.
Why is the volume of a cone one third of a cylinder?
A cone fits inside a cylinder with the same base and height, and it holds exactly a third of it: fill the cone with water three times and you fill the cylinder. The same third turns up for a pyramid inside a prism, so V = (1/3)Bh covers both.
Which geometry formulas should I memorize?
Fewer than the sheet suggests. Learn the rectangle, the triangle (half a rectangle), the circle, V = Bh for anything with straight sides and the same cross section, and the one third for anything that comes to a point. Most of the rest are those with a particular base put in.
What units do area and volume use?
Lengths and perimeters are in plain units (cm), areas in square units (cm²) and volumes in cubic units (cm³). Checking the units is the fastest way to catch a wrong formula: if a volume came out in cm², a length was left out.
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