Geometry Basics
Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. From the angles in a triangle to the volume of a cylinder, this guide covers the core concepts you need for middle and high school math.
Types of Angles
An angle is formed when two rays (lines that start at a point and go in one direction) meet at a common endpoint called the vertex. Angles are measured in degrees (written with the symbol °). A full rotation is 360°.
| Angle Type | Measure | Description |
|---|---|---|
| Acute | 0° – 89° | Less than a right angle; looks "sharp" |
| Right | 90° | A perfect corner; marked with a small square |
| Obtuse | 91° – 179° | Greater than a right angle; looks "wide" |
| Straight | 180° | A straight line |
| Reflex | 181° – 359° | Greater than a straight angle; the "outside" angle |
ACUTE (<90) RIGHT (=90) OBTUSE (>90) STRAIGHT (180)
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/ | \
/ | \
Supplementary and Complementary Angles
Complementary angles add up to 90°. If one angle is 35°, its complement is 55°.
Supplementary angles add up to 180°. If one angle is 110°, its supplement is 70°.
Triangles
A triangle is a three-sided polygon. The angles inside any triangle always add up to exactly 180°. Triangles are classified two ways: by their sides and by their angles.
Classification by Sides
SCALENE ISOSCELES EQUILATERAL
(all sides (2 sides (all sides
different) equal) equal)
/\ /\ /\
/ \ / \ / \
/ \ / \ / \
/______\ /______\ /________\
a b c a b a a a a
| Type | Sides | Angles |
|---|---|---|
| Scalene | All different lengths | All different |
| Isosceles | Two equal sides | Two equal base angles |
| Equilateral | All three equal | All three = 60° |
| Right | Any | One angle = exactly 90° |
A triangle has angles of 47° and 63°. What is the third angle?
180 – 47 – 63 = 70°
Quadrilaterals
A quadrilateral is any four-sided polygon. The four interior angles of any quadrilateral always sum to 360°.
| Shape | Properties |
|---|---|
| Square | 4 equal sides, 4 right angles |
| Rectangle | Opposite sides equal, 4 right angles |
| Parallelogram | Opposite sides equal and parallel; no right angles required |
| Rhombus | 4 equal sides, opposite angles equal (like a tilted square) |
| Trapezoid | Exactly one pair of parallel sides |
Perimeter
The perimeter of a shape is the total distance around its outside edges. Add up the lengths of all sides.
A rectangle 8 cm long and 5 cm wide:
P = 2(8) + 2(5) = 16 + 10 = 26 cm
Sides of 7, 9, and 12 inches:
P = 7 + 9 + 12 = 28 inches
Area Formulas
Area measures the amount of surface inside a shape. It is always given in square units (cm², m², in², etc.).
| Shape | Formula | Variables |
|---|---|---|
| Rectangle / Square | A = l × w | l = length, w = width |
| Triangle | A = (b × h) / 2 | b = base, h = height |
| Parallelogram | A = b × h | b = base, h = perpendicular height |
| Trapezoid | A = (b₁ + b₂) / 2 × h | b₁, b₂ = parallel sides, h = height |
| Circle | A = πr² | r = radius (π ≈ 3.14159) |
Base = 10 cm, Height = 6 cm
A = (10 × 6) / 2 = 60 / 2 = 30 cm²
Radius = 7 m
A = π × 7² = 3.14159 × 49 ≈ 153.94 m²
Circumference of a Circle
The circumference is the perimeter of a circle — the distance all the way around it.
Two equivalent formulas: C = 2πr or C = πd, where r is the radius and d is the diameter (d = 2r).
A circle has a diameter of 10 cm.
C = π × 10 ≈ 3.14159 × 10 ≈ 31.42 cm
The Pythagorean Theorem
The Pythagorean Theorem applies only to right triangles. It states that the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
a² + b² = c²
|\
| \
b | \ c (hypotenuse)
| \
|____\
a
A right triangle has legs of 3 and 4. Find the hypotenuse.
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25 = 5
The 3-4-5 triangle is the most famous Pythagorean triple.
Hypotenuse = 13, one leg = 5. Find the other leg.
a² + 5² = 13²
a² + 25 = 169
a² = 144
a = 12
Common Pythagorean Triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 6 | 8 | 10 |
Volume
Volume measures the amount of three-dimensional space inside a solid. It is given in cubic units (cm³, m³, in³).
Rectangular Prism (Box)
V = l × w × h
V = 6 × 4 × 3 = 72 cm³
Cylinder
V = πr²h (the area of the circular base times the height)
V = π × 5² × 10 = 3.14159 × 25 × 10 ≈ 785.4 cm³
Formula Reference Table
| Shape | Perimeter / Circumference | Area | Volume |
|---|---|---|---|
| Rectangle | 2l + 2w | l × w | — |
| Triangle | a + b + c | bh / 2 | — |
| Circle | 2πr or πd | πr² | — |
| Rect. Prism | — | — | l × w × h |
| Cylinder | — | — | πr²h |
Frequently Asked Questions
What is the difference between area and perimeter?
Perimeter is the total length of the boundary around a shape — it is a one-dimensional measurement (just a length). Area is the amount of flat surface inside a shape — it is a two-dimensional measurement given in square units. Think of it this way: perimeter is the length of fence you need to enclose a yard, while area is the amount of grass inside that fence.
Can the Pythagorean Theorem be used on any triangle?
No. The Pythagorean Theorem only works on right triangles — triangles that contain exactly one 90° angle. For other triangles, you need different tools like the Law of Sines or Law of Cosines (topics covered in trigonometry). If you apply a² + b² = c² to a non-right triangle, the equation will not be true.
What is pi and why do we use it for circles?
Pi (π) is the ratio of any circle's circumference to its diameter. No matter how large or small the circle, dividing its circumference by its diameter always gives the same number: approximately 3.14159... The digits of π continue forever without repeating, making it an irrational number. We use π in circle formulas because it captures the fundamental relationship between a circle's dimensions that holds true for every circle in existence.
How do I find the area of an irregular shape?
Break the irregular shape into simpler shapes whose area formulas you know (rectangles, triangles, semicircles, etc.). Calculate the area of each piece separately, then add them all together. For example, an L-shaped room can be split into two rectangles. A shape with a semicircular cutout would have the semicircle area subtracted from the rectangle area. This "decomposition" strategy works for virtually any polygon.
Quick Quiz
Check your understanding. Click an answer to see if you got it right.