Chapter 7: Geometry — Area, Perimeter & Volume
Geometry brings algebra to life by connecting formulas to real shapes and spaces. In this chapter you will learn the vocabulary of basic geometry, classify angles and triangles, and master formulas for perimeter, area, surface area, and volume. We close with the Pythagorean theorem — one of the most celebrated results in all of mathematics.
7.1 Basic Geometry Vocabulary
Fundamental Terms
- Point: An exact location in space, with no size. Labeled with a capital letter (e.g., point $A$).
- Line: A straight path extending infinitely in both directions. Written $\overleftrightarrow{AB}$.
- Line Segment: Part of a line with two endpoints. Written $\overline{AB}$; length = $AB$.
- Ray: Starts at a point and extends infinitely in one direction. Written $\overrightarrow{AB}$.
- Plane: A flat surface extending infinitely in all directions.
- Angle: Formed by two rays sharing a common endpoint (the vertex). Measured in degrees ($^\circ$).
Types of Angles
| Angle Type | Measure | Symbol / Description |
|---|---|---|
| Acute | $0^\circ < \theta < 90^\circ$ | Sharp angle |
| Right | $\theta = 90^\circ$ | Square corner; marked with a small square |
| Obtuse | $90^\circ < \theta < 180^\circ$ | Wider than a right angle |
| Straight | $\theta = 180^\circ$ | A line |
| Reflex | $180^\circ < \theta < 360^\circ$ | Greater than a straight angle |
Angle Relationships
- Complementary angles: Two angles whose measures sum to $90^\circ$.
- Supplementary angles: Two angles whose measures sum to $180^\circ$.
- Vertical angles: Opposite angles formed by two intersecting lines; they are equal.
7.2 Triangles
Classification by Sides
- Equilateral: All three sides equal; all angles $= 60^\circ$.
- Isosceles: Exactly two sides equal; the angles opposite the equal sides are equal.
- Scalene: All three sides different lengths.
Classification by Angles
- Acute triangle: All angles less than $90^\circ$.
- Right triangle: One angle exactly $90^\circ$.
- Obtuse triangle: One angle greater than $90^\circ$.
Triangle Angle Sum Theorem
The sum of the interior angles of any triangle is always $180^\circ$:
$$\angle A + \angle B + \angle C = 180^\circ$$Example 1 — Finding a Missing Angle
A triangle has angles $47^\circ$ and $68^\circ$. Find the third angle.
$$\angle C = 180^\circ - 47^\circ - 68^\circ = 65^\circ$$Since all angles are less than $90^\circ$, this is an acute triangle.
7.3 Perimeter of Polygons
The perimeter is the total distance around the outside of a shape. Add all side lengths.
| Shape | Perimeter Formula | Variables |
|---|---|---|
| Rectangle | $P = 2l + 2w$ | $l$ = length, $w$ = width |
| Square | $P = 4s$ | $s$ = side length |
| Triangle | $P = a + b + c$ | $a, b, c$ = three side lengths |
| Regular $n$-gon | $P = ns$ | $n$ = number of sides, $s$ = side length |
| Circle (circumference) | $C = 2\pi r = \pi d$ | $r$ = radius, $d$ = diameter |
7.4 Area Formulas
Area measures the amount of flat surface enclosed by a shape. Area is always measured in square units (cm², m², ft², etc.).
| Shape | Area Formula | Notes |
|---|---|---|
| Rectangle | $A = lw$ | length $\times$ width |
| Square | $A = s^2$ | side squared |
| Triangle | $A = \dfrac{1}{2}bh$ | $b$ = base, $h$ = perpendicular height |
| Parallelogram | $A = bh$ | base $\times$ perpendicular height |
| Trapezoid | $A = \dfrac{1}{2}(b_1 + b_2)h$ | average of two bases $\times$ height |
| Circle | $A = \pi r^2$ | $r$ = radius; $\pi \approx 3.14159$ |
Example 2 — Area of a Trapezoid
A trapezoid has parallel bases of $8$ cm and $14$ cm, and a height of $5$ cm. Find its area.
$$A = \frac{1}{2}(b_1 + b_2)h = \frac{1}{2}(8 + 14)(5) = \frac{1}{2}(22)(5) = 55 \text{ cm}^2$$Example 3 — Area of a Circle
A circular garden has a diameter of $10$ m. Find its area. Use $\pi \approx 3.14$.
7.5 Surface Area and Volume
Surface area (SA) is the total area of all faces of a 3-D solid, measured in square units. Volume (V) is the amount of space inside a solid, measured in cubic units.
| Solid | Surface Area | Volume |
|---|---|---|
| Rectangular Prism | $SA = 2(lw + lh + wh)$ | $V = lwh$ |
| Cube | $SA = 6s^2$ | $V = s^3$ |
| Cylinder | $SA = 2\pi r^2 + 2\pi r h$ | $V = \pi r^2 h$ |
| Triangular Prism | $SA = bh + (a+b+c)H$ | $V = \dfrac{1}{2}bh \cdot H$ |
| Cone | $SA = \pi r^2 + \pi r l$ ($l$ = slant height) | $V = \dfrac{1}{3}\pi r^2 h$ |
| Sphere | $SA = 4\pi r^2$ | $V = \dfrac{4}{3}\pi r^3$ |
Example 4 — Volume of a Rectangular Prism
A box is $6$ ft long, $4$ ft wide, and $3$ ft tall. Find its volume and surface area.
$$V = lwh = 6 \times 4 \times 3 = 72 \text{ ft}^3$$ $$SA = 2(lw + lh + wh) = 2(24 + 18 + 12) = 2(54) = 108 \text{ ft}^2$$Example 5 — Volume of a Cylinder
A cylindrical can has radius $3$ cm and height $10$ cm. Find its volume. Use $\pi \approx 3.14$.
$$V = \pi r^2 h \approx 3.14 \times 9 \times 10 = 282.6 \text{ cm}^3$$7.6 The Pythagorean Theorem
Pythagorean Theorem
In any right triangle, with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle):
$$a^2 + b^2 = c^2$$The hypotenuse is always the longest side and is always opposite the right angle ($90^\circ$).
The theorem has two primary uses:
- Find the hypotenuse: $c = \sqrt{a^2 + b^2}$
- Find a missing leg: $a = \sqrt{c^2 - b^2}$
Example 6 — Finding the Hypotenuse
A right triangle has legs $a = 5$ and $b = 12$. Find the hypotenuse.
$$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$The 5-12-13 triple is a common Pythagorean triple.
Example 7 — Real-World Application
A 10-foot ladder leans against a wall. Its base is 6 feet from the wall. How high up the wall does it reach?
Common Pythagorean Triples to Memorize
These sets of whole numbers satisfy $a^2 + b^2 = c^2$:
- $3, 4, 5$ (and multiples: $6, 8, 10$; $9, 12, 15$)
- $5, 12, 13$
- $8, 15, 17$
- $7, 24, 25$
Practice Problems
Practice — Chapter 7
- Two angles are supplementary. One measures $73^\circ$. Find the other.
- A triangle has angles $90^\circ$ and $35^\circ$. Find the third angle and classify the triangle.
- Find the perimeter of a rectangle with $l = 9$ m and $w = 5$ m.
- Find the area of a triangle with base $14$ cm and height $6$ cm.
- A circle has radius $7$ in. Find (a) the circumference and (b) the area. Use $\pi \approx 3.14$.
- Find the volume of a rectangular prism with dimensions $5 \times 3 \times 4$ cm.
- A cylinder has diameter $8$ ft and height $5$ ft. Find its volume. Use $\pi \approx 3.14$.
- A right triangle has legs $9$ and $40$. Find the hypotenuse.
- Is a triangle with sides $10$, $24$, $26$ a right triangle? Justify using the Pythagorean theorem.
Show Answers
- $180^\circ - 73^\circ = 107^\circ$
- $180^\circ - 90^\circ - 35^\circ = 55^\circ$; right triangle (has a $90^\circ$ angle).
- $P = 2(9)+2(5) = 28$ m
- $A = \frac{1}{2}(14)(6) = 42$ cm$^2$
- (a) $C = 2(3.14)(7) = 43.96$ in; (b) $A = 3.14(49) = 153.86$ in$^2$
- $V = 5 \times 3 \times 4 = 60$ cm$^3$
- $r = 4$; $V = 3.14(16)(5) = 251.2$ ft$^3$
- $c = \sqrt{81+1600} = \sqrt{1681} = 41$
- $10^2 + 24^2 = 100 + 576 = 676 = 26^2$ ✓ Yes, it is a right triangle.
Chapter Summary
- Geometry vocabulary: points, lines, rays, planes, angles, and polygons.
- Angle types: acute ($<90^\circ$), right ($=90^\circ$), obtuse ($>90^\circ$); complementary pairs sum to $90^\circ$; supplementary pairs sum to $180^\circ$.
- The three interior angles of any triangle sum to $180^\circ$.
- Perimeter = sum of all sides; Area = flat surface enclosed (in square units).
- Key area formulas: rectangle $lw$, triangle $\frac{1}{2}bh$, trapezoid $\frac{1}{2}(b_1+b_2)h$, circle $\pi r^2$.
- Volume measures 3-D space (cubic units); surface area measures all outer faces (square units).
- Pythagorean Theorem: $a^2 + b^2 = c^2$ for right triangles, where $c$ is the hypotenuse.