Essential Geometry Formulas for Math Tests

In This Guide

Perimeter formulas
Area formulas
Circle formulas
Pythagorean theorem
Distance and midpoint
Surface area
Volume formulas
Similarity and scale factor
Right-triangle trigonometry
Common formula mistakes

How to Use Geometry Formulas Correctly

Before choosing a formula, identify the figure, label the known measurements, and state what the problem asks you to find. A diagram may contain more information than the formula needs. For example, the area of a triangle requires a base and its corresponding perpendicular height, not just any two side lengths.

1. Identify the figure

Decide whether the problem involves a triangle, rectangle, circle, prism, cylinder, cone, sphere, or coordinate-plane figure.

2. Identify the requested quantity

Perimeter, area, surface area, volume, distance, midpoint, missing side, or angle each requires a different relationship.

3. Match the units

Convert measurements to the same unit before substituting into a formula.

4. Check the dimension

Lengths use linear units, areas use square units, and volumes use cubic units.

Formula selection rule

Do not start by asking which formula you remember. Start by asking what geometric quantity the problem describes and which measurements define that quantity.

1. Perimeter Formulas

Perimeter is the total distance around a two-dimensional figure. Add the lengths of all outer sides. Perimeter is measured in linear units such as centimeters, meters, inches, or feet.

Rectangle perimeter

P = 2 l + 2 w

Here, l is length and w is width. The formula may also be written as:

P = 2 (l + w)

Square perimeter

P = 4 s

General polygon perimeter

P = s_{1} + s_{2} + \dots + s_{n}

Example: Rectangle perimeter

A rectangular garden is 14 meters long and 9 meters wide. Find its perimeter.

P = 2 (14 + 9) = 2 (23) = 46

Answer

The perimeter is 46 meters.

2. Area Formulas

Area measures the amount of two-dimensional space inside a figure. Answers use square units because area is based on two dimensions.

Rectangle area

A = l w

Square area

A = s^{2}

Triangle area

A = \frac{1}{2} b h

The height must be perpendicular to the chosen base. In an obtuse triangle, the perpendicular height may fall outside the triangle.

Parallelogram area

A = b h

Trapezoid area

A = \frac{1}{2} (b_{1} + b_{2}) h

Example: Triangle area

A triangle has a base of 12 centimeters and a perpendicular height of 7 centimeters. Find its area.

A = \frac{1}{2} (12) (7) = 42

Answer

The area is 42 square centimeters.

3. Circle Formulas

A circle is defined by its center and radius. The diameter is twice the radius.

d = 2 r

Circumference

C = 2 π r

Using the diameter:

C = π d

Circle area

A = π r^{2}

Example: Circumference and area

A circle has radius 5 inches. Find its circumference and area in exact form.

C = 2 π (5) = 10 π

A = π {(5)}^{2} = 25 π

Answer

The circumference is 10π inches, and the area is 25π square inches.

Radius and diameter error

If the problem gives a diameter, divide by 2 before using the area formula. Substituting the diameter for r makes the calculated area four times too large.

4. The Pythagorean Theorem

The Pythagorean theorem applies only to right triangles. The hypotenuse is the side opposite the right angle and is always the longest side.

a^{2} + b^{2} = c^{2}

Here, c is the hypotenuse. The theorem can be rearranged to find a missing leg:

a = \sqrt{c^{2} - b^{2}}

Example: Finding the hypotenuse

A right triangle has legs of length 6 and 8. Find the hypotenuse.

6^{2} + 8^{2} = c^{2}

36 + 64 = c^{2} ⟹ 100 = c^{2} ⟹ c = 10

Answer

The hypotenuse is 10 units.

5. Coordinate Geometry Formulas

Distance formula

The distance formula is the Pythagorean theorem applied to horizontal and vertical changes between two coordinate points.

d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}

Midpoint formula

M (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Example: Distance and midpoint

Find the distance and midpoint between A(1, 2) and B(7, 10).

d = \sqrt{{(7 - 1)}^{2} + {(10 - 2)}^{2}} = \sqrt{36 + 64} = 10

M (\frac{1 + 7}{2}, \frac{2 + 10}{2}) = (4, 6)

Answer

The distance is 10 units, and the midpoint is (4, 6).

6. Surface Area Formulas

Surface area is the total area of all outside faces or curved surfaces of a three-dimensional object. Answers use square units.

Rectangular prism

SA = 2 l w + 2 l h + 2 w h

Cube

SA = 6 s^{2}

Cylinder

SA = 2 π r^{2} + 2 π r h

Sphere

SA = 4 π r^{2}

Open and closed solids

A container without a lid has less surface area than a closed prism. Read the description carefully and include only the surfaces that actually exist.

7. Volume Formulas

Volume measures the amount of three-dimensional space inside a solid. Answers use cubic units.

Rectangular prism

V = l w h

General prism

V = B h

Here, B is the area of the base.

Cylinder

V = π r^{2} h

Pyramid

V = \frac{1}{3} B h

Cone

V = \frac{1}{3} π r^{2} h

Sphere

V = \frac{4}{3} π r^{3}

Example: Cylinder volume

A cylinder has radius 3 centimeters and height 10 centimeters. Find its exact volume.

V = π {(3)}^{2} (10) = 90 π

Answer

The volume is 90π cubic centimeters.

8. Similarity and Scale Factor

Similar figures have equal corresponding angles and proportional corresponding side lengths. The scale factor compares a length in the new figure with the corresponding length in the original figure.

k = \frac{new length}{original length}

Lengths scale by k, areas scale by k squared, and volumes scale by k cubed.

\frac{new area}{original area} = k^{2}

\frac{new volume}{original volume} = k^{3}

Example: Area under a scale factor

A figure is enlarged by a scale factor of 3. The original area is 8 square units. Find the new area.

New area = 8 {(3)}^{2} = 8 (9) = 72

Answer

The new area is 72 square units.

9. Right-Triangle Trigonometry

Sine, cosine, and tangent relate an acute angle in a right triangle to ratios of side lengths.

\sin θ = \frac{opposite}{hypotenuse}

\cos θ = \frac{adjacent}{hypotenuse}

\tan θ = \frac{opposite}{adjacent}

The words opposite and adjacent depend on the selected reference angle. The hypotenuse remains the side opposite the right angle.

Example: Finding a missing side

In a right triangle, an acute angle is 35 degrees and the adjacent side is 12 units. Find the opposite side to the nearest tenth.

\tan (35 °) = \frac{x}{12}

x = 12 \tan (35 °) \approx 8.4

Answer

The opposite side is approximately 8.4 units.

10. Interior Angle Sum of a Polygon

The sum of the interior angles of an n-sided polygon is:

S = (n - 2) \times 180 °

For a regular polygon, divide the total by n to find each interior angle:

Each interior angle = \frac{(n - 2) \times 180 °}{n}

Example: Regular octagon

Find the measure of each interior angle of a regular octagon.

\frac{(8 - 2) \times 180 °}{8} = \frac{1080 °}{8} = 135 °

Answer

Each interior angle measures 135 degrees.

Geometry Units at a Glance

Quantity	Typical units	Example
Length, distance, perimeter, circumference	Linear units	12 cm, 5 m, 8 ft
Area and surface area	Square units	24 cm², 10 m²
Volume	Cubic units	90 cm³, 18 ft³
Angles	Degrees or radians	45°, π/3 radians
Slope	Vertical units per horizontal unit	5 meters per second

Common Geometry Formula Mistakes

Using diameter as the radius

Area and most circle formulas use r. Divide the diameter by 2 before substituting.

Using a slanted side as height

Height must be perpendicular to the base in triangle, parallelogram, trapezoid, prism, pyramid, and cone formulas.

Forgetting square or cubic units

Area uses square units; volume uses cubic units. A numerical answer without the correct unit is incomplete.

Using the Pythagorean theorem on any triangle

The theorem applies only when the triangle contains a right angle.

Confusing perimeter with area

Perimeter measures the boundary; area measures the region inside the boundary.

Using length scale factor for area

If lengths scale by k, areas scale by k² and volumes by k³.

Rounding too early

Keep π and radical values exact until the final step unless the problem asks for a decimal estimate.

Including hidden or missing surfaces

Surface area depends on whether the solid is open, closed, joined to another object, or missing a face.

A Fast Geometry Formula Checklist

What figure or solid is shown?
What quantity must be found?
Which measurements are given?
Are all measurements in compatible units?
Is the height perpendicular to the base?
Is the circle measurement a radius or diameter?
Is the triangle a right triangle?
Should the answer use linear, square, or cubic units?
Should π or radicals remain exact?
Does the result make sense compared with the dimensions?

How to Remember Geometry Formulas

Memorizing a formula is easier when you understand its structure. Rectangle area is length times width because it counts rows and columns of square units. Triangle area is half of a matching parallelogram. Prism volume is base area times height because equal cross-sections are stacked through the solid.

Group formulas by purpose instead of memorizing one long list: boundary formulas, two-dimensional area formulas, right-triangle relationships, coordinate formulas, surface area, volume, similarity, and trigonometry. Then practice identifying which family a problem belongs to before calculating.

Practice Geometry Questions

Apply these formulas to triangles, circles, coordinate geometry, area, perimeter, surface area, volume, trigonometry, and mixed exam-style questions.

All Geometry Questions Coordinate Geometry Trigonometry Questions SAT Geometry and Trigonometry Mixed Exam-Style Practice