Geometry Formula Sheet: Essential Formulas With Explanations

Perimeter and circumference

Perimeter is the total distance around a polygon. Add all side lengths. For a rectangle, $P=2l+2w$. For a square, $P=4s$.

For a circle, circumference is $C=2\pi r$ or $C=\pi d$, where $r$ is radius and $d$ is diameter. Circumference uses linear units.

• Rectangle: $P=2l+2w$
• Square: $P=4s$
• Triangle: $P=a+b+c$
• Circle: $C=2\pi r=\pi d$

Area of common plane figures

Area measures the space inside a two-dimensional figure and uses square units. The height in triangle and parallelogram formulas must be perpendicular to the base.

For composite figures, divide the diagram into familiar shapes, find each area, and add or subtract as needed.

• Rectangle: $A=lw$
• Square: $A=s^2$
• Triangle: $A=\dfrac{1}{2}bh$
• Parallelogram: $A=bh$
• Trapezoid: $A=\dfrac{1}{2}(b_1+b_2)h$
• Circle: $A=\pi r^2$

Surface area

Surface area is the total area covering the outside of a three-dimensional solid, so answers use square units. A net can make the faces easier to count.

For prisms, $2B$ represents the two congruent bases and $Ph$ represents the rectangular lateral faces, where $B$ is base area and $P$ is base perimeter.

• Rectangular prism: $SA=2lw+2lh+2wh$
• Right prism: $SA=2B+Ph$
• Cylinder: $SA=2\pi r^2+2\pi rh$
• Sphere: $SA=4\pi r^2$
• Cone: $SA=\pi r^2+\pi r\ell$

Volume

Volume measures three-dimensional space and uses cubic units. For any prism or cylinder, multiply base area by perpendicular height.

Pyramids and cones have one-third the volume of a prism or cylinder with the same base and height.

• Prism: $V=Bh$
• Rectangular prism: $V=lwh$
• Cylinder: $V=\pi r^2h$
• Pyramid: $V=\dfrac{1}{3}Bh$
• Cone: $V=\dfrac{1}{3}\pi r^2h$
• Sphere: $V=\dfrac{4}{3}\pi r^3$

Right triangles and coordinate geometry

The Pythagorean theorem, $a^2+b^2=c^2$, applies only to right triangles; $c$ is the hypotenuse opposite the right angle. Trigonometric ratios connect an acute angle to side lengths.

On the coordinate plane, subtraction finds horizontal and vertical change. Keep point order consistent in slope calculations.

• Pythagorean theorem: $a^2+b^2=c^2$
• Slope: $m=\dfrac{y_2-y_1}{x_2-x_1}$
• Distance: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
• Midpoint: $\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$
• $\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$
• $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$

Angle relationships and polygon facts

Complementary angles sum to $90^\circ$, supplementary angles sum to $180^\circ$, and vertical angles are congruent. A triangle’s interior angles sum to $180^\circ$.

The interior-angle sum of an $n$-sided polygon is $(n-2)180^\circ$. If the polygon is regular, divide that result by $n$ for each interior angle.

• Triangle sum: $180^\circ$
• Quadrilateral sum: $360^\circ$
• Polygon interior sum: $(n-2)180^\circ$
• Regular polygon interior angle: $\dfrac{(n-2)180^\circ}{n}$
• Exterior angles, one at each vertex: sum to $360^\circ$

How to use a formula sheet accurately

Draw or mark the diagram, write the formula in symbols, substitute with parentheses, and attach units. Estimate before calculating so a misplaced radius or exponent is easier to catch.

For circles, check whether the given length is a radius or diameter. For solids, distinguish slant height from perpendicular height.