Geometry: Cheat Sheet

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Essential Concepts

  • Geometry is the study of shapes and their properties. In geometry, we use points, lines, line segments, and rays to describe different things.
  • Angles are formed when two lines, line segments, or rays meet at a point. They can be measured in degrees or radians using a protractor. There are five types of angles: right angle ([latex]90[/latex] degrees, like an “L” shape), acute angle (less than [latex]90[/latex] degrees, smaller than a right angle), obtuse angle (more than [latex]90[/latex] degrees but less than [latex]180[/latex] degrees, larger than a right angle), straight angle ([latex]180[/latex] degrees, forms a straight line), and reflex angle (more than [latex]180[/latex] degrees but less than [latex]360[/latex] degrees, larger than a straight angle).
  • Measuring angles involves using a tool called a protractor. To measure an angle, place the protractor on the vertex (endpoint) of the angle and align the straight edge with one side. Read the degree value where the other side intersects the protractor.
  • Angles can be measured in degrees or radians. Degrees are commonly used in everyday situations like navigation and construction, while radians are used in advanced mathematics and scientific fields. The conversion factor between degrees and radians is [latex]\text{Angle in Degrees} = \text{Angle in Radians }\times\frac{180}{π}[/latex].
  • The measure of an angle represents the amount of rotation or separation between two lines or line segments. It is denoted by “m” followed by the angle symbol, such as “[latex]m\angle[/latex].”
  • When two angles add up to [latex]180[/latex] degrees, they are called supplementary angles, and when they add up to [latex]90[/latex] degrees, they are called complementary angles. These concepts can be represented as equations, where the sum of the measures of the angles equals [latex]180[/latex] degrees for supplementary angles and [latex]90[/latex] degrees for complementary angles.
  • In geometry, we use formulas and a problem-solving strategy to solve geometry-related problems by understanding the given information, identifying what we are looking for, translating it into equations, solving the equations, and checking our answer to ensure it makes sense.
  • When two lines intersect, the opposite angles formed are called vertical angles. Vertical angles have the same measure, meaning they are equal.
  • When a line crosses two or more other lines, it is called a transversal. This creates different angles:
    • Alternate interior angles are on opposite sides of the transversal and inside the other lines and they have the same measure.
    • Alternate exterior angles are on opposite sides of the transversal and outside the other lines and they also have the same measure.
    • Corresponding angles are in the same position with respect to the transversal but on different lines and they have equal measures too.
  • Triangles can be classified based on their side lengths and angle measures.
    • By side lengths:
      • An equilateral triangle has all sides of equal length and all angles measure [latex]60[/latex] degrees.
      • An isosceles triangle has two sides of equal length and two angles are equal.
      • A scalene triangle has no sides of equal length and all angles are different.
    • By angle measures:
      • A right triangle has one angle that measures exactly [latex]90[/latex] degrees.
      • An acute triangle has all three angles measuring less than [latex]90[/latex] degrees.
      • An obtuse triangle has one angle that measures more than [latex]90[/latex] degrees.
  • The sum of the measures of the angles of a triangle is always [latex]180[/latex] degrees.
  • Similar triangles are figures that have the same shape but different sizes. They are like scale models of each other. When two triangles are similar, their corresponding sides have the same ratio and their corresponding angles have the same measure. This means that if one side of a triangle is four times longer than the corresponding side of another triangle, all the other sides will also have that same ratio, and the angles will be equal as well.
  • The Pythagorean theorem states that in a right triangle (a triangle with one [latex]90[/latex]-degree angle), the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In other words, if we label the lengths of the sides as “[latex]a[/latex],” “[latex]b[/latex],” and “[latex]c[/latex],” with “[latex]c[/latex]” being the hypotenuse, the theorem can be written as [latex]{a}^{2}+{b}^{2}={c}^{2}[/latex].
  • Rectangles have four sides with four right angles. Opposite sides of a rectangle are equal in length. The perimeter of a rectangle is found by adding twice the length and twice the width. The perimeter formula is [latex]P = 2L + 2W[/latex]. The area of a rectangle is calculated by multiplying the length and width together. The area formula is [latex]A = LW[/latex]
  • Triangles are polygons with three sides and three angles. The sum of the interior angles of any triangle is always [latex]180[/latex] degrees. The perimeter of a triangle is the sum of the lengths of the sides. The area of a triangle can be found by multiplying the base length and the height and dividing it by [latex]2[/latex].
  • A trapezoid is a quadrilateral with four sides. It has two parallel sides and two non-parallel sides. The area of a trapezoid can be calculated by multiplying the height of the trapezoid by the sum of the lengths of the two parallel sides, and then dividing the result by [latex]2[/latex].
  • A circle is a shape with a curved boundary and no straight sides. The radius ([latex]r[/latex]) is the distance from the center of the circle to any point on its boundary, and the diameter ([latex]d[/latex]) is the distance across the circle passing through the center. The diameter is always twice the length of the radius ([latex]d = 2r[/latex]). The circumference of a circle is the perimeter, and it can be found by multiplying the diameter by [latex]\pi[/latex] (pi) ([latex]C=2\pi r[/latex]). The area of a circle is found by multiplying [latex]\pi[/latex] (pi) by the square of the radius ([latex]A=\pi {r}^{2}[/latex]).
  • A rectangular solid, also known as a rectangular prism, is a three-dimensional shape with six rectangular faces. Here are the properties of a rectangular solid:
    • Volume: The volume of a rectangular solid is the amount of space inside it. To find the volume, multiply the length ([latex]L[/latex]), width ([latex]W[/latex]), and height ([latex]H[/latex]) of the solid together: [latex]Volume = L × W × H[/latex].
    • Surface Area: The surface area of a rectangular solid is the sum of the areas of all its faces. To find the surface area, add up the areas of all six faces: [latex]\mbox{Surface Area} = 2(LW + LH + WH)[/latex], where [latex]LW[/latex] represents the area of the front and back faces, [latex]LH[/latex] represents the area of the top and bottom faces, and [latex]WH[/latex] represents the area of the two side faces.
  • A sphere is a three-dimensional shape that is perfectly round and symmetrical. Here are the properties of a sphere:
    • Volume: The volume of a sphere is the amount of space enclosed by it. To find the volume, use the formula: [latex]Volume=\Large\frac{4}{3}\normalsize\pi {r}^{3}[/latex], where “[latex]r[/latex]” represents the radius of the sphere.
    • Surface Area: The surface area of a sphere is the total area of its curved surface. To find the surface area, use the formula: [latex]\mbox{Surface Area}=4\pi {r}^{2}[/latex], where “[latex]r[/latex]” represents the radius of the sphere.
  • A cylinder is a three-dimensional shape with two circular bases that are parallel and congruent. Here are the properties of a cylinder:
    • Volume: The volume of a cylinder is the amount of space inside it. To find the volume, multiply the area of the base (A) by the height (h) of the cylinder: [latex]Volume = A × h[/latex]. For a cylinder, the base area is given by the formula [latex]A=\pi {r}^{2}[/latex], where “[latex]r[/latex]” represents the radius of the base.
    • Surface Area: The surface area of a cylinder is the sum of the areas of its curved surface (lateral surface) and the areas of its two circular bases. To find the surface area, use the formula: [latex]\mbox{Surface Area}=2\pi {r}^{2}+2\pi rh[/latex], where “[latex]r[/latex]” represents the radius of the base and “[latex]h[/latex]” represents the height of the cylinder.
  • A cone is a three-dimensional shape with a circular base that tapers to a single point called the apex. Here are the properties of a cone:
    • Volume: The volume of a cone is the amount of space inside it. To find the volume, multiply the area of the base ([latex]A[/latex]) by the height ([latex]h[/latex]) of the cone and divide it by [latex]3[/latex]: [latex]Volume = \Large\frac{4}{3} × A × h[/latex]. For a cone, the base area is given by the formula [latex]A=\pi {r}^{2}[/latex], where “[latex]r[/latex]” represents the radius of the base.

Glossary

angle

formed when two lines, line segments, or rays meet at a common endpoint called the vertex

circumference

the distance around a circle

diameter

a line segment that passes through a circle’s center connecting two points on the circle

line

an infinite collection of points extending infinitely in both directions

line segment

a finite portion of a line with two endpoints

measure of an angle

a numerical value that represents the amount of rotation or separation between two lines or line segments

parallel lines

lines that lie in the same plane and move in the same direction, but never intersect

perpendicular lines

two lines that intersect at a [latex]90^\circ[/latex] angle

plane

a two-dimensional surface with infinite length and width, and no thickness

point

a location in space with no length, width, or height

radius

a line segment from the center to any point on the circle

ray

a part of a line that starts at an endpoint and extends infinitely in one direction

surface area

a square measure of the total area of all the sides

transversal

a line that intersects two or more other parallel lines creating eight angles, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles

vertical angles

a pair of opposite angles that are formed when two lines intersect

volume

a cubic measure of the amount of space inside an object

Key Equations

area

[latex]A=L\cdot W[/latex]

area of a circle

[latex]A=\pi {r}^{2}[/latex]

area of a triangle

[latex]A={\Large\frac{1}{2}}bh[/latex]

[latex]{\text{Area}}_{\text{trapezoid}}[/latex]

[latex]\text{A}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex]

circumference

[latex]C=2\pi r[/latex]

conversion factor between degrees and radians

[latex]\text{Angle in Degrees} = \text{Angle in Radians }\times\frac{180}{π}[/latex], [latex]\text{Angle in Radians} = \text{Angle in Degrees }\times\frac{π}{180}[/latex]

perimeter

[latex]P=2L+2W \text{ or } P = 2(L+W)[/latex]

perimeter of a triangle

[latex]P=a+b+c[/latex]

sum of the measures of the angles of a triangle

[latex]m\angle A+m\angle B+m\angle C=\text{180}^ \circ[/latex]

surface area of a cube

[latex]S=6{s}^{2}[/latex]

surface area of a cylinder

[latex]S=2\pi {r}^{2}+2\pi rh[/latex]

surface area of a sphere

[latex]S=4\pi {r}^{2}[/latex]

The Pythagorean Theorem

[latex]{a}^{2}+{b}^{2}={c}^{2}[/latex]

volume of a cone

[latex]\frac{1}{3}\pi{r}^{2}h[/latex]

volume of a cube

[latex]V={s}^{3}[/latex]

volume of a cylinder

[latex]\pi {r}^{2}h[/latex]

volume of a sphere

[latex]V=\Large\frac{4}{3}\normalsize\pi {r}^{3}[/latex]