Area and volume

Explanation

Area measures two-dimensional surface space (in square units like $\text{cm}^2$ ), while volume measures the three-dimensional space an object occupies (in cubic units like $\text{cm}^3$ ).

Formulas for 2D Shapes and 3D Solids

2D Area Formulas

Shape	Formula	Variable Definitions
Rectangle / Parallelogram	$\text{Area} = b \cdot h$	$b = \text{base}$ , $h = \text{height}$
Triangle	$\text{Area} = \frac{1}{2}b \cdot h$	$b = \text{base}$ , $h = \text{height}$
Trapezoid	$\text{Area} = \frac{1}{2}(b_1 + b_2)h$	$b_1, b_2 = \text{parallel bases}$ , $h = \text{height}$
Circle	$\text{Area} = \pi r^2$	$r = \text{radius}$

3D Volume Formulas

Solid	Formula	Variable Definitions
Rectangular Prism	$\text{Volume} = l \cdot w \cdot h$	$l = \text{length}$ , $w = \text{width}$ , $h = \text{height}$
Right Cylinder	$\text{Volume} = \pi r^2 h$	$r = \text{radius}$ , $h = \text{height}$
Right Cone	$\text{Volume} = \frac{1}{3}\pi r^2 h$	$r = \text{radius}$ , $h = \text{height}$
Sphere	$\text{Volume} = \frac{4}{3}\pi r^3$	$r = \text{radius}$
Square Pyramid	$\text{Volume} = \frac{1}{3}l \cdot w \cdot h$	$l = \text{base length}$ , $w = \text{base width}$ , $h = \text{height}$

Example 1

What is the length of one side of a square that has the same area as a circle with radius

3

3

3\pi

3\sqrt{\pi}

9\pi

Choice C is correct. The area of a circle with radius

r

A=\pi r^2

, so a circle with radius

3

has area

\pi(3)^2=9\pi

. The area of a square with side length

s

A=s^2

, so if the square has the same area as the circle, then

s^2=9\pi

. Since a side length must be positive, taking the square root of both sides gives

s=\sqrt{9\pi}=\sqrt{9}\cdot\sqrt{\pi}=3\sqrt{\pi}

. Choice A is incorrect; the side length of the square is not equal to the radius of the circle. Choice B is incorrect and may result from writing

\sqrt{9\pi}

3\pi

instead of correctly simplifying the square root. Choice D is incorrect; this is the area of the circle,

9\pi

, not the side length of the square.

Example 2

The circumference of the base of a right circular cylinder is

16\pi

meters, and the height of the cylinder is

5

meters. What is the volume, in cubic meters, of the cylinder?

40\pi

80\pi

320\pi

1{,}280\pi

Choice C is correct. The volume of a right circular cylinder is

V=\pi r^2 h

, where

r

is the radius of the base and

h

is the height. The circumference of the base is

C=2\pi r

, so setting

2\pi r=16\pi

and dividing both sides by

2\pi

gives

r=8

. Substituting

r=8

and

h=5

into the volume formula gives

V=\pi(8)^2(5)=\pi(64)(5)=320\pi

. Choice A is incorrect and may result from a conceptual or calculation error in finding the radius. Choice B is incorrect; this results from using

r=4

(half of the correct radius) in the volume formula. Choice D is incorrect; this results from using the circumference itself,

16

, as the radius instead of solving for the actual radius,

8

Tips for solving

Understand how scaling side lengths affects area versus volume.
Example: A rectangle has its dimensions doubled ( $k = 2$ ). How do area and volume scale?
- Length / Perimeter: Scales by $k \implies$ Doubles ( $2 \times$ )
- Area: Scales by $k^2 \implies$ Quadruples ( $2^2 = 4 \times$ )
- Volume: Scales by $k^3 \implies$ Octuples ( $2^3 = 8 \times$ )
Solve composite area problems by subtracting the inner shape from the outer shape.
Example: Find the shaded area of a square with side length $10\text{ cm}$ containing a circle of radius $4\text{ cm}$ inside it.
- Outer Square Area: $10^2 = 100\text{ cm}^2$
- Inner Circle Area: $\pi (4^2) = 16\pi\text{ cm}^2$
- Shaded Region Area: $\text{Square} - \text{Circle} = 100 - 16\pi \approx 49.73\text{ cm}^2$
Relate cylinder volume to cone volume when they share the same base radius and height.
Example: A cylinder and a cone both have radius $r = 5\text{ cm}$ and height $h = 12\text{ cm}$ .
- Cylinder Volume: $\pi r^2 h = \pi(5^2)(12) = 300\pi\text{ cm}^3$
- Cone Volume: $\frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(5^2)(12) = 100\pi\text{ cm}^3$
Watch out for diameter vs. radius in SAT geometric word problems.
Example: A cylindrical tank has a diameter of $10\text{ ft}$ and height of $8\text{ ft}$ . Calculate volume.
- Find Radius First: Divide diameter by $2 \implies r = \frac{10}{2} = 5\text{ ft}$
- Calculate Volume: $\text{Volume} = \pi r^2 h = \pi(5^2)(8) = 200\pi\text{ ft}^3$

Practice questions

00:00

Question 1 · easy

A circle has a radius of

6

meters. What is the area, in square meters, of the circle?

Question 2 · easy

A cube has an edge length of

5

inches. What is the volume, in cubic inches, of the cube?

Question 3 · easy

A right circular cylinder has a radius of

3

feet and a height of

10

feet. What is the volume, in cubic feet, of the cylinder?

Question 4 · easy

A closed rectangular box has length

8

inches, width

5

inches, and height

4

inches. What is the total exterior surface area, in square inches, of the box?

Question 5 · easy

A circle has a diameter of

10

meters. What is the area, in square meters, of the circle?

Question 6 · easy

The length of each edge of a box is

13

inches. Each side of the box is a square, and the box does not have a lid. What is the exterior surface area, in square inches, of this box without a lid?

Question 7 · easy

A solid sphere has a radius of

6

centimeters. What is the volume, in cubic centimeters, of the sphere?

Question 8 · easy

A rectangular prism has length

7

centimeters, width

4

centimeters, and height

3

centimeters. What is the volume, in cubic centimeters, of the prism?

Question 9 · easy

A cube has an edge length of

9

centimeters. What is the total exterior surface area, in square centimeters, of the cube?

Question 10 · easy

A right square prism has a base edge length of

6

units and a height of

11

units. What is the volume, in cubic units, of the prism?

SAT Prep