Geometry is the branch of mathematics that studies shape, size, position, distance, direction, and space. It helps us describe everything from a single point to a complex building, a road map, a football field, or the orbit of a satellite.
In geometry, we ask questions such as: How long is this side? What angle do these lines make? How much space is inside this shape? Are these two figures the same shape? Where is a point located? The answers come from clear definitions, diagrams, measurements, formulas, and logical reasoning.
This guide builds geometry from the ground up. You will meet the basic objects, learn how angles and shapes are classified, understand the difference between two-dimensional and three-dimensional figures, use important formulas, and work through examples. You do not need to memorize everything at once. Focus first on what each idea means, then practise choosing the right relationship for the problem.
Why do we learn geometry?
Geometry connects visual thinking with precise reasoning. A carpenter checks lengths and right angles. An architect plans floor area and three-dimensional space. A game designer places objects on coordinate grids. A scientist uses models of circles, spheres, and paths. Students also use geometry in algebra, trigonometry, calculus, physics, engineering, art, design, maps, and computer graphics.
The subject develops three useful habits:
- Seeing structure: breaking a complicated figure into familiar triangles, rectangles, circles, or solids.
- Explaining why: supporting an answer with a definition, a known fact, or a logical chain rather than guessing from appearance.
- Checking reasonableness: noticing whether a length, angle, area, or volume makes sense in the context.
The basic language of geometry
Geometry begins with a few ideal objects. A drawing represents these objects, but the mathematical idea is more exact than any pencil mark.
Point
A point marks an exact location. It has no length, width, or thickness. Points are usually named with capital letters such as \(A\), \(B\), and \(C\).
Line
A line is straight, has no thickness, and continues forever in both directions. The line through points \(A\) and \(B\) can be written as \(\overleftrightarrow{AB}\).
Line segment
A line segment is the part of a line between two endpoints. Segment \(AB\), written \(\overline{AB}\), has a measurable length.
Ray
A ray begins at one endpoint and continues forever in one direction. In \(\overrightarrow{AB}\), \(A\) is the endpoint and the ray passes through \(B\).
Planes and space
A plane is a flat surface that extends forever in every direction. A sheet of paper suggests a plane, although real paper has edges and thickness. A floor, wall, or screen can also help you picture part of a plane. Three-dimensional space contains points, lines, planes, and solid figures.
Collinear, coplanar, and intersection
- Points on the same line are collinear.
- Points or lines on the same plane are coplanar.
- When geometric objects meet, their shared point or set of points is their intersection.
Two different lines in a plane may intersect once or never intersect. Lines in the same plane that never meet are parallel. Lines that meet to make four right angles are perpendicular. In three-dimensional space, two lines can also be skew: they do not meet and are not parallel because they lie in different planes.
Angles: measuring a turn
An angle is formed by two rays with a common endpoint. The common endpoint is the vertex, and the rays are the sides of the angle. Angles are commonly measured in degrees. One complete turn is \(360^\circ\), a half-turn is \(180^\circ\), and a quarter-turn is \(90^\circ\).
| Angle type | Measure | How to recognize it |
|---|---|---|
| Acute | \(0^\circ \lt \theta \lt 90^\circ\) | Smaller than a right angle |
| Right | \(\theta=90^\circ\) | Usually marked with a small square |
| Obtuse | \(90^\circ \lt \theta \lt 180^\circ\) | Larger than a right angle but smaller than a straight angle |
| Straight | \(\theta=180^\circ\) | Its sides form a straight line |
| Reflex | \(180^\circ \lt \theta \lt 360^\circ\) | The larger turn around the vertex |
Angle relationships
Some angle pairs are useful because their measures are connected:
- Complementary angles add to \(90^\circ\).
- Supplementary angles add to \(180^\circ\).
- Vertical angles are opposite angles made by two intersecting lines, and they are equal.
- A linear pair consists of adjacent angles whose non-shared sides form a straight line, so they add to \(180^\circ\).
Worked example: angles on a straight line
Two adjacent angles form a straight line. One measures \(127^\circ\). Let the other measure be \(x^\circ\).
Subtract \(127\) from both sides:
The missing angle is \(53^\circ\). A quick check gives \(53^\circ+127^\circ=180^\circ\).
Polygons and their properties
A polygon is a closed, flat figure made only from straight line segments. The segments are its sides, their meeting points are vertices, and the inside turns are interior angles. A circle is not a polygon because its boundary is curved. A figure with a gap is not a polygon because it is not closed.
| Number of sides | Name | Familiar example |
|---|---|---|
| 3 | Triangle | A triangular road sign |
| 4 | Quadrilateral | A rectangle or square |
| 5 | Pentagon | A five-sided badge |
| 6 | Hexagon | A cell in a honeycomb |
| 8 | Octagon | A common stop-sign outline |
| \(n\) | \(n\)-gon | A general polygon with \(n\) sides |
Regular and irregular polygons
A regular polygon has all sides equal and all interior angles equal. A regular triangle is equilateral, and a regular quadrilateral is a square. An irregular polygon does not satisfy both conditions. A rectangle with unequal adjacent sides is irregular as a polygon even though it is highly symmetrical.
The interior-angle sum of a polygon
From one vertex of a convex \(n\)-sided polygon, draw diagonals to every non-adjacent vertex. The polygon splits into \(n-2\) triangles. Because every triangle has angle sum \(180^\circ\), the polygon’s interior-angle sum is:
Worked example: interior angles of a regular hexagon
A hexagon has \(n=6\) sides, so its interior-angle sum is:
In a regular hexagon, all six angles are equal. Each angle is therefore:
Triangles: geometry’s building blocks
A triangle has three sides, three vertices, and three interior angles. Its interior angles always add to \(180^\circ\). Triangles are especially important because many complex figures can be divided into triangles.
Classifying triangles by sides
- An equilateral triangle has three equal sides and three \(60^\circ\) angles.
- An isosceles triangle has at least two equal sides. The angles opposite those equal sides are equal.
- A scalene triangle has three different side lengths.
Classifying triangles by angles
- An acute triangle has three acute angles.
- A right triangle has one \(90^\circ\) angle.
- An obtuse triangle has one obtuse angle.
Quadrilaterals
A quadrilateral is any polygon with four sides. Its interior angles add to \(360^\circ\). The names below describe properties, and one shape can belong to more than one group.
- A parallelogram has two pairs of opposite parallel sides.
- A rectangle is a parallelogram with four right angles.
- A rhombus is a parallelogram with four equal sides.
- A square has four equal sides and four right angles, so it is a rectangle, a rhombus, and a parallelogram.
- A trapezoid has at least one pair of parallel sides under the inclusive definition commonly used in mathematics. Some courses use an exclusive definition requiring exactly one pair, so follow your course’s stated convention.
- A kite has two pairs of adjacent equal sides.
Congruence, similarity, and symmetry
Congruent figures
Two figures are congruent when they have the same shape and the same size. One can be moved, turned, or reflected to fit exactly on the other. Corresponding sides and corresponding angles are equal. The symbol for congruence is \(\cong\).
Similar figures
Two figures are similar when they have the same shape but may have different sizes. Corresponding angles are equal, and corresponding side lengths have one common scale factor. If every length in a small figure is multiplied by \(k\), the new figure has perimeter multiplied by \(k\), area multiplied by \(k^2\), and volume multiplied by \(k^3\).
Worked example: a scale drawing
A drawing uses a scale of \(1:50\). A wall measures \(8\text{ cm}\) in the drawing. Its real length is:
The units matter. Multiplying by the scale gives centimetres first, so the result must then be converted to metres.
Transformations and symmetry
A translation slides a figure, a rotation turns it around a centre, and a reflection flips it across a line. These three transformations preserve lengths and angles, so the image is congruent to the original. A dilation enlarges or reduces a figure from a centre using a scale factor. It preserves angle measures and creates a similar image.
A figure has line symmetry if reflection across a line maps it onto itself. It has rotational symmetry if a turn smaller than \(360^\circ\) maps it onto itself. For example, a non-square rectangle has two lines of symmetry and rotational symmetry of order \(2\).
Definitions, theorems, and proof
Geometry is not only a collection of formulas. It is also a system for showing why a statement must be true. A definition explains exactly what a term means. A theorem is a statement established by logical reasoning. A proof connects known facts in a sequence where every conclusion follows from an earlier step.
Suppose two parallel lines are crossed by a third line. The resulting angle relationships are not accepted because the picture looks convincing; they follow from established properties of parallel lines. Once one angle is known, vertical angles, linear pairs, and corresponding angles can determine the others. Each reason matters as much as each number.
A simple reasoning chain
If two intersecting lines make one angle of \(68^\circ\), its vertical angle is also \(68^\circ\). Each adjacent angle forms a linear pair with it, so each adjacent angle measures:
The four angles are therefore \(68^\circ\), \(112^\circ\), \(68^\circ\), and \(112^\circ\). Their total is \(360^\circ\), which provides a useful check.
When writing a proof, name the objects clearly, state the fact being used, and avoid relying on visual appearance. A diagram can guide your thinking, but the reasoning makes the conclusion dependable—even if the sketch is not drawn to scale.
Perimeter, area, and circumference
Perimeter is the total distance around a closed two-dimensional figure. Area measures the amount of surface inside it. The circumference is the perimeter of a circle. These quantities use different units: perimeter uses linear units such as centimetres, while area uses square units such as square centimetres.
| Shape | Perimeter or circumference | Area |
|---|---|---|
| Rectangle with length \(l\) and width \(w\) | \(P=2l+2w\) | \(A=lw\) |
| Square with side \(s\) | \(P=4s\) | \(A=s^2\) |
| Triangle with sides \(a,b,c\), base \(b\), height \(h\) | \(P=a+b+c\) | \(A=\frac{1}{2}bh\) |
| Parallelogram with base \(b\), side \(a\), height \(h\) | \(P=2a+2b\) | \(A=bh\) |
| Trapezoid with parallel sides \(a,b\) and height \(h\) | Add all four sides | \(A=\frac{1}{2}(a+b)h\) |
| Circle with radius \(r\) and diameter \(d\) | \(C=2\pi r=\pi d\) | \(A=\pi r^2\) |
The height in an area formula means the perpendicular distance from the base to the opposite side or vertex. A sloping side is not automatically the height. This is one of the most common sources of errors in triangle, parallelogram, and trapezoid questions.
Worked example: rectangle
A rectangular study desk is \(1.2\text{ m}\) long and \(0.6\text{ m}\) wide.
The perimeter is a distance, so it uses metres. The surface area uses square metres.
Worked example: triangle
A triangle has a base of \(14\text{ cm}\) and a perpendicular height of \(9\text{ cm}\).
Only the base and its corresponding perpendicular height are needed. The other side lengths are unnecessary for this area calculation.
Circles
A circle is the set of all points in a plane that are the same distance from a fixed point called the centre. That fixed distance is the radius, \(r\). A diameter passes through the centre and has both endpoints on the circle, so \(d=2r\). A chord joins any two points on the circle, while an arc is part of the curved boundary.
The number \(\pi\) is the constant ratio of a circle’s circumference to its diameter. It is approximately \(3.14159\), but keep the \(\pi\) symbol during a calculation when an exact answer is useful.
Worked example: circle
A circular garden has radius \(5\text{ m}\). Its circumference and area are:
Notice that the radius is squared only in the area formula. Students sometimes square it when finding circumference, which produces the wrong type of unit.
If you want a guided tool for checking two-dimensional and three-dimensional measurements, use the area, volume, and surface area calculator. Use a calculator to check arithmetic after you have chosen the formula, not as a substitute for deciding what the problem is asking.
The Pythagorean theorem
In a right triangle, the side opposite the right angle is the hypotenuse. It is the longest side. If the shorter legs have lengths \(a\) and \(b\), and the hypotenuse has length \(c\), then:
This relationship works only for right triangles. It can find a missing side, test whether three side lengths form a right triangle, or calculate a straight-line distance from horizontal and vertical changes.
Worked example: finding the hypotenuse
A right triangle has legs \(6\text{ cm}\) and \(8\text{ cm}\). Let the hypotenuse be \(c\).
Length is positive, so the relevant square root is \(10\), not \(-10\).
Worked example: finding a shorter side
A right triangle has hypotenuse \(13\text{ m}\) and one leg \(5\text{ m}\). Let the missing leg be \(x\).
Coordinate geometry
Coordinate geometry joins algebra and geometry. On a coordinate plane, a point is written as an ordered pair \((x,y)\). The \(x\)-coordinate gives horizontal position and the \(y\)-coordinate gives vertical position. The axes meet at the origin, \((0,0)\).
Distance between two points
For points \((x_1,y_1)\) and \((x_2,y_2)\), the horizontal change is \(x_2-x_1\) and the vertical change is \(y_2-y_1\). These changes form the legs of a right triangle, so the Pythagorean theorem gives:
Midpoint and slope
The midpoint averages the two \(x\)-coordinates and the two \(y\)-coordinates:
The slope measures vertical change divided by horizontal change:
A horizontal line has slope \(0\). A vertical line has undefined slope because its horizontal change is \(0\), and division by zero is undefined.
Worked example: distance and midpoint
Let \(A=(-1,2)\) and \(B=(5,10)\). The coordinate changes are \(6\) horizontally and \(8\) vertically.
The segment has length \(10\) units and midpoint \((2,6)\).
Three-dimensional geometry
Two-dimensional figures have length and width. Three-dimensional solids also have height or depth, so they occupy space. Polyhedra are solids with flat polygonal faces. Curved solids such as cylinders, cones, and spheres are not polyhedra.
- A face is a flat surface of a polyhedron.
- An edge is a line segment where two faces meet.
- A vertex is a point where edges meet.
For every convex polyhedron, Euler’s relationship connects the numbers of vertices \(V\), edges \(E\), and faces \(F\):
A cube has \(V=8\), \(E=12\), and \(F=6\), so \(8-12+6=2\). This is a useful counting check, but it does not apply in the same way to solids with curved surfaces.
| Solid | Volume | Surface area |
|---|---|---|
| Rectangular prism, length \(l\), width \(w\), height \(h\) | \(V=lwh\) | \(S=2lw+2lh+2wh\) |
| Cube, side \(s\) | \(V=s^3\) | \(S=6s^2\) |
| Cylinder, radius \(r\), height \(h\) | \(V=\pi r^2h\) | \(S=2\pi r^2+2\pi rh\) |
| Cone, radius \(r\), height \(h\) | \(V=\frac{1}{3}\pi r^2h\) | \(S=\pi r^2+\pi r\ell\), where \(\ell\) is slant height |
| Sphere, radius \(r\) | \(V=\frac{4}{3}\pi r^3\) | \(S=4\pi r^2\) |
Volume uses cubic units because it counts three-dimensional space. Surface area uses square units because it measures the combined area of the outside surfaces. For visual explanations of common solids, visit the guide to 3D shape names and properties.
Worked example: cylinder
A closed cylinder has radius \(3\text{ cm}\) and height \(10\text{ cm}\).
The surface area includes two circular ends and the curved side. If the cylinder were an open container, the missing top would have to be removed from the calculation.
How to solve a geometry problem
- Read for the target. Decide whether the question asks for a length, angle, perimeter, area, surface area, volume, proof, or classification.
- Sketch and label. Draw a clear figure if one is not provided. Mark known lengths, angle measures, parallel lines, right angles, and equal parts.
- Translate words into relationships. For example, “supplementary” means the angles add to \(180^\circ\), while “radius” means half the diameter.
- Choose a formula or theorem. Check that its conditions are satisfied. The Pythagorean theorem, for instance, requires a right triangle.
- Substitute with units. Keep exact values such as \(\pi\) or square roots until the final step unless the question asks for a decimal.
- Solve carefully. Show enough algebra that another student could follow your reasoning.
- Check the result. Confirm the unit, sign, size, and fit with the diagram. An area cannot be negative, and a triangle’s angles must total \(180^\circ\).
Common mistakes and how to avoid them
Using the wrong kind of measurement
Perimeter is around, area is inside a flat figure, surface area is the outside of a solid, and volume is the space inside a solid. Write the requested quantity beside your work before selecting a formula.
Forgetting squared or cubed units
If lengths are in centimetres, area is in \(\text{cm}^2\) and volume is in \(\text{cm}^3\). The exponent belongs to the unit as well as to the calculation.
Trusting the drawing too much
A sketch may not be to scale. Do not assume lines are parallel, sides are equal, or an angle is \(90^\circ\) unless the problem states or marks it, or you can prove it.
Rounding too early
Keep \(\pi\), fractions, and square roots during intermediate steps. Round once at the end to the requested precision. Early rounding can make a final answer noticeably inaccurate.
Mixing units
Convert all measurements to one compatible unit before substituting. For example, do not multiply a length in metres by a width in centimetres without converting first.
Check your understanding
- Two angles are complementary. One is \(38^\circ\). What is the other?
- Find the interior-angle sum of a decagon.
- A rectangle measures \(11\text{ cm}\) by \(7\text{ cm}\). Find its perimeter and area.
- A right triangle has legs \(9\text{ cm}\) and \(12\text{ cm}\). Find its hypotenuse.
- Find the midpoint of \((-4,3)\) and \((8,9)\).
- A cube has side length \(4\text{ cm}\). Find its surface area and volume.
1. Complementary angle
\(90^\circ-38^\circ=52^\circ\).
2. Decagon angle sum
A decagon has \(n=10\), so \((10-2)\times180^\circ=1440^\circ\).
3. Rectangle
\(P=2(11)+2(7)=36\text{ cm}\), and \(A=(11)(7)=77\text{ cm}^2\).
4. Right triangle
\(c=\sqrt{9^2+12^2}=\sqrt{225}=15\text{ cm}\).
5. Midpoint
\(M=\left(\frac{-4+8}{2},\frac{3+9}{2}\right)=(2,6)\).
6. Cube
\(S=6(4^2)=96\text{ cm}^2\), and \(V=4^3=64\text{ cm}^3\).
How to improve at geometry
Geometry becomes easier when diagrams, words, and formulas are studied together. Redraw figures neatly, label only verified information, and explain each step in a short phrase. Practise mixed questions so that you learn to choose a method instead of merely copying the most recent formula.
For extra practice, use the geometry worksheets with answer sheets. Students preparing for a course assessment can also review the structured Geometry EOC study guide. When checking a solution, ask not only “Is the final number correct?” but also “Why was this relationship valid?”
A compact geometry summary
- Points describe locations; lines, segments, and rays describe straight paths.
- Angles measure turns and are classified by their degree measures.
- Polygons are closed plane figures made from straight sides.
- Congruent figures have the same shape and size; similar figures have the same shape with proportional lengths.
- Perimeter and circumference measure distance around; area measures flat surface; volume measures three-dimensional space.
- Coordinate geometry expresses shapes and distances using numbers and algebra.
- A correct solution uses definitions and conditions, not the appearance of the diagram alone.