39 formulas

Geometry Formula Sheet

These are the formulas a high school geometry course uses: area and perimeter, circles, triangles and angles, right triangles and trig ratios, the coordinate plane, volume, and surface area.

Each formula shows what its letters stand for. Open any one for a worked example and a calculator, or print the sheet for test review.

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Area and perimeter

Area of a Rectangle
$A = lw$
Calculates the area of a rectangle given length and width.
$l$
Length of the rectangle (units)
$w$
Width of the rectangle (units)
Worked example and calculator
Perimeter of a Rectangle
$P = 2(l + w)$
Calculates the perimeter of a rectangle.
$l$
Length of the rectangle (units)
$w$
Width of the rectangle (units)
Worked example and calculator
Area of a Square
$A = s^2$
Calculates the area of a square given side length.
$s$
Length of one side (units)
Worked example and calculator
Perimeter of a Square
$P = 4s$
Calculates the perimeter of a square.
$s$
Length of one side (units)
Worked example and calculator
Area of a Triangle
$A = \frac{1}{2}bh$
Calculates the area of a triangle given its base and height.
$b$
Length of the base (units)
$h$
Perpendicular height from base to opposite vertex (units)
Worked example and calculator
Area of a Parallelogram
$A = bh$
Calculates the area of a parallelogram given base and height.
$b$
Length of the base (units)
$h$
Perpendicular height (units)
Worked example and calculator
Area of a Trapezoid
$A = \frac{1}{2}(b_1 + b_2)h$
Calculates the area of a trapezoid given parallel sides and height.
$b_1$
Length of first parallel side (units)
$b_2$
Length of second parallel side (units)
$h$
Perpendicular height between bases (units)
Worked example and calculator
Area of a Rhombus or Kite
$A = \frac{1}{2} d_1 d_2$
Area of a rhombus or kite equals half the product of its two diagonals d₁ and d₂.
$d_1$
Length of one diagonal
$d_2$
Length of the other diagonal
Worked example and calculator
Area of a Regular Polygon
$A = \frac{1}{2} a P$
Area of a regular polygon equals half the apothem a times the perimeter P.
$a$
Distance from the center to the midpoint of a side
$P$
Total distance around the polygon
Worked example and calculator

Circles

Area of a Circle
$A = \pi r^2$
Calculates the area enclosed by a circle with radius r.
$r$
Distance from center to edge (units)
Worked example and calculator
Circumference of a Circle
$C = 2\pi r$
Calculates the perimeter (distance around) a circle.
$r$
Distance from center to edge (units)
Worked example and calculator
Arc Length
$s = \frac{\theta}{360^\circ}\cdot 2\pi r$
The length of a circular arc that spans a central angle θ (in degrees) in a circle of radius r.
$\theta$
The angle the arc subtends at the center (°)
$r$
Radius of the circle
Worked example and calculator
Area of a Sector
$A = \frac{\theta}{360^\circ}\cdot \pi r^2$
The area of a pie-slice sector that spans a central angle θ (in degrees) in a circle of radius r.
$\theta$
The angle of the sector at the center (°)
$r$
Radius of the circle
Worked example and calculator
Circle (Standard Form)
$(x - h)^2 + (y - k)^2 = r^2$
A circle with center (h, k) and radius r: all points at fixed distance r from the center.
$h$
x-coordinate of center
$k$
y-coordinate of center
$r$
Radius length (units)
$x$
Point x-coordinate
$y$
Point y-coordinate
Worked example and calculator

Triangles and angles

Triangle Angle Sum
$A + B + C = 180^\circ$
The three interior angles of any triangle always add up to 180 degrees.
$A$
A known interior angle (°)
$B$
A second known interior angle (°)
Worked example and calculator
Triangle Inequality
$a + b > c, \quad a + c > b, \quad b + c > a$
Three lengths form a triangle only if each side is shorter than the sum of the other two.
$a$
First length
$b$
Second length
$c$
Third length
Worked example and calculator
Similar Triangles (Proportional Sides)
$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$
Similar triangles have equal angles and proportional corresponding sides; set up a proportion to find a missing length.
$a$
A side of the first triangle
$a'$
The corresponding side of the second triangle
Worked example and calculator
Polygon Interior Angle Sum
$S = (n - 2)\cdot 180^\circ$
The interior angles of an n-sided polygon add up to (n − 2) × 180 degrees.
$n$
How many sides (and vertices) the polygon has
Worked example and calculator
Regular Polygon Angles
$\text{Interior} = \frac{(n - 2)\cdot 180^\circ}{n}, \quad \text{Exterior} = \frac{360^\circ}{n}$
Each interior and exterior angle of a regular polygon (all sides and angles equal) with n sides.
$n$
Sides of the regular polygon
Worked example and calculator

Right triangles and trigonometry

Pythagorean Theorem
$a^2 + b^2 = c^2$
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This is one of the most important formulas in all of mathematics.
$a$
One of the two shorter sides of the right triangle, the sides that form the right angle (units)
$b$
The other shorter side of the right triangle, together with leg a, they form the 90° corner (units)
$c$
The longest side, always opposite the right angle, this is the side you solve for most often (units)
Worked example and calculator
Special Right Triangles
$45\text{-}45\text{-}90:\ 1 : 1 : \sqrt{2} \qquad 30\text{-}60\text{-}90:\ 1 : \sqrt{3} : 2$
Fixed side ratios for the two special right triangles, so you can find sides without trigonometry.
$x$
The shorter leg (or either leg in a 45-45-90)
Worked example and calculator
Sine (SOH)
$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
In a right triangle, sine of an angle equals the opposite side divided by the hypotenuse.
$\theta$
The angle (degrees or radians)
$opp$
Side opposite the angle (units)
$hyp$
Longest side of the right triangle (units)
Worked example and calculator
Cosine (CAH)
$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
In a right triangle, cosine of an angle equals the adjacent side divided by the hypotenuse.
$\theta$
The angle (degrees or radians)
$adj$
Side next to the angle (not the hypotenuse) (units)
$hyp$
Longest side of the right triangle (units)
Worked example and calculator
Tangent (TOA)
$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$
In a right triangle, tangent of an angle equals the opposite side divided by the adjacent side.
$\theta$
The angle (degrees or radians)
$opp$
Side opposite the angle (units)
$adj$
Side next to the angle (not the hypotenuse) (units)
Worked example and calculator

Coordinate geometry

Distance Formula
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Calculates the straight-line distance between two points on a coordinate plane.
$x_1$
X-coordinate of first point
$y_1$
Y-coordinate of first point
$x_2$
X-coordinate of second point
$y_2$
Y-coordinate of second point
Worked example and calculator
Midpoint Formula
$M = \left(\frac{x_1 + x_2}{2},\, \frac{y_1 + y_2}{2}\right)$
Finds the exact middle point between two coordinates.
$x_1$
X-coordinate of first point
$y_1$
Y-coordinate of first point
$x_2$
X-coordinate of second point
$y_2$
Y-coordinate of second point
Worked example and calculator
Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Calculates the slope (steepness) of a line between two points.
$x_1$
X-coordinate of first point
$y_1$
Y-coordinate of first point
$x_2$
X-coordinate of second point
$y_2$
Y-coordinate of second point
Worked example and calculator

Volume

Volume of a Cube
$V = s^3$
Calculates the volume of a cube given its side length.
$s$
Edge length of the cube (units)
Worked example and calculator
Volume of a Rectangular Prism
$V = lwh$
Calculates the volume of a rectangular prism (box).
$l$
Length of the prism (units)
$w$
Width of the prism (units)
$h$
Height of the prism (units)
Worked example and calculator
Volume of a Cylinder
$V = \pi r^2 h$
Calculates the volume of a cylinder given its radius and height.
$r$
Radius of the circular base (units)
$h$
Height of the cylinder (units)
Worked example and calculator
Volume of a Pyramid
$V = \frac{1}{3}Bh$
Calculates the volume of a pyramid.
$B$
Area of the base (units²)
$h$
Height from base to apex (units)
Worked example and calculator
Volume of a Cone
$V = \frac{1}{3}\pi r^2 h$
Calculates the volume of a cone.
$r$
Radius of the circular base (units)
$h$
Height of the cone (units)
Worked example and calculator
Volume of a Sphere
$V = \frac{4}{3}\pi r^3$
Calculates the volume of a sphere.
$r$
Radius of the sphere (units)
Worked example and calculator

Surface area

Surface Area of a Cube
$SA = 6 s^2$
Total surface area of a cube with edge length s (six equal square faces).
$s$
Length of one edge of the cube
Worked example and calculator
Surface Area of a Rectangular Prism
$A = 2(lw + lh + wh)$
Calculates the total surface area of a rectangular prism.
$l$
Length of the prism (units)
$w$
Width of the prism (units)
$h$
Height of the prism (units)
Worked example and calculator
Surface Area of a Cylinder
$SA = 2\pi r^2 + 2\pi r h$
Total surface area of a closed cylinder: two circular ends plus the curved side.
$r$
Radius of the circular base
$h$
Height of the cylinder
Worked example and calculator
Surface Area of a Square Pyramid
$SA = b^2 + 2 b l$
Surface area of a pyramid with a square base of side b and slant height l: the base plus four triangular faces.
$b$
Side length of the square base
$l$
Height of a triangular face from base edge to apex
Worked example and calculator
Surface Area of a Cone
$A = \pi r(r + l)$
Calculates the total surface area of a cone.
$r$
Radius of the circular base (units)
$l$
Distance from base to apex along the surface (units)
Worked example and calculator
Surface Area of a Sphere
$A = 4\pi r^2$
Calculates the surface area of a sphere.
$r$
Radius of the sphere (units)
Worked example and calculator

Learn these formulas

Other formula sheets

Geometry Formula Sheet

These are the formulas a high school geometry course uses: area and perimeter, circles, triangles and angles, right triangles and trig ratios, the coordinate plane, volume, and surface area.

Each formula shows what its letters stand for. Open any one for a worked example and a calculator, or print the sheet for test review.

Area and perimeter

Area of a Rectangle

A = lw

Calculates the area of a rectangle given length and width.

$l$: Length of the rectangle (units)
$w$: Width of the rectangle (units)

Area and perimeter

Area of a Rectangle

Perimeter of a Rectangle

Area of a Square

Perimeter of a Square

Area of a Triangle

Area of a Parallelogram

Area of a Trapezoid

Area of a Rhombus or Kite

Area of a Regular Polygon

Circles

Area of a Circle

Circumference of a Circle

Arc Length

Area of a Sector

Circle (Standard Form)

Triangles and angles

Triangle Angle Sum

Triangle Inequality

Similar Triangles (Proportional Sides)

Polygon Interior Angle Sum

Regular Polygon Angles

Right triangles and trigonometry

Pythagorean Theorem

Special Right Triangles

Sine (SOH)

Cosine (CAH)

Tangent (TOA)

Coordinate geometry

Distance Formula

Midpoint Formula

Slope Formula

Volume

Volume of a Cube

Volume of a Rectangular Prism

Volume of a Cylinder

Volume of a Pyramid

Volume of a Cone

Volume of a Sphere

Surface area

Surface Area of a Cube

Surface Area of a Rectangular Prism

Surface Area of a Cylinder

Surface Area of a Square Pyramid

Surface Area of a Cone

Surface Area of a Sphere

Learn these formulas

Other formula sheets

Area and perimeter

Area of a Rectangle

Perimeter of a Rectangle

Area of a Square

Perimeter of a Square

Area of a Triangle

Area of a Parallelogram

Area of a Trapezoid

Area of a Rhombus or Kite

Area of a Regular Polygon

Circles

Area of a Circle

Circumference of a Circle

Arc Length

Area of a Sector

Circle (Standard Form)

Triangles and angles

Triangle Angle Sum

Triangle Inequality

Similar Triangles (Proportional Sides)

Polygon Interior Angle Sum

Regular Polygon Angles

Right triangles and trigonometry

Pythagorean Theorem

Special Right Triangles

Sine (SOH)

Cosine (CAH)

Tangent (TOA)

Coordinate geometry

Distance Formula

Midpoint Formula

Slope Formula