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Using Formulas
Area & Volume

Learn to use formulas for area and volume, understand what each variable represents, and solve real-world problems with confidence.

Pre-Algebra

What Is a Formula?

A formula is a mathematical rule written with variables and operations that shows the relationship between quantities. Instead of starting from scratch every time, you plug in values and calculate the answer.

Why Do Formulas Matter?

Efficiency
One formula works for millions of different shapes: just change the numbers.
Real World
Builders, engineers, and designers use area and volume formulas every day.
Precision
Formulas give exact answers: no guessing, no estimating.

Area vs Volume: The Big Picture

2D Surface
Area
measured in \( \text{units}^2 \)
vs
3D Space
Volume
measured in \( \text{units}^3 \)

Core Area Formulas

Rectangle
\( A = l \times w \)
\( l \) = length, \( w \) = width
Triangle
\( A = \frac{1}{2} \times b \times h \)
\( b \) = base, \( h \) = height
Circle
\( A = \pi r^2 \)
\( r \) = radius, \( \pi \approx 3.14 \)

Core Volume Formulas

Rectangular Prism
\( V = l \times w \times h \)
\( l \) = length, \( w \) = width, \( h \) = height
Cylinder
\( V = \pi r^2 h \)
\( r \) = radius of base, \( h \) = height
Triangular Prism
\( V = \frac{1}{2} \times b \times h_t \times l \)
\( b \) = base of triangle, \( h_t \) = triangle height, \( l \) = length

Visualizing the Variables

\( l = 9 \text{ cm} \)
\( w = 6 \text{ cm} \)
\( A = 54 \text{ cm}^2 \)
Rectangle
\( l = 5 \)
\( w = 3 \)
\( h = 4 \)
\( V = 60 \text{ units}^3 \)
Rectangular Prism

Square Units vs Cubic Units

Area Volume
Measures Flat surface covered Space inside a 3D shape
Dimensions 2 (length \( \times \) width) 3 (length \( \times \) width \( \times \) height)
Units \( \text{cm}^2, \text{m}^2, \text{in}^2, \text{ft}^2 \) \( \text{cm}^3, \text{m}^3, \text{in}^3, \text{ft}^3 \)
Example Paint for a wall Water in a pool

How to Apply a Formula

1
Identify the shape. Is it a rectangle, triangle, circle, prism, or cylinder?
2
Choose the correct formula. Match the shape to its area or volume formula.
3
List the known values. Write down each variable and what number it equals.
4
Substitute and compute. Replace variables with numbers, then simplify step by step.
5
Label your answer. Include the correct units (squared for area, cubed for volume).

Example 1: Rectangle Area

A garden is 12 m long and 7 m wide. Find the area.

\( l = 12 \text{ m} \)
\( w = 7 \text{ m} \)
\( A = ? \)

Example 2: Triangle Area

A triangular flag has a base of 10 in and a height of 6 in. Find the area.

\( b = 10 \text{ in} \)
\( h = 6 \text{ in} \)

Example 3: Rectangular Prism Volume

A fish tank is 20 in long, 10 in wide, and 12 in tall. Find the volume.

\( l = 20 \text{ in} \)
\( w = 10 \text{ in} \)
\( h = 12 \text{ in} \)

Example 4: Circle Area

A circular pizza has a radius of 7 in. Find the area. Use \( \pi \approx 3.14 \).

\( r = 7 \text{ in} \)
center

Example 5: Cylinder Volume

A can has a radius of 4 cm and a height of 10 cm. Find the volume. Use \( \pi \approx 3.14 \).

\( r = 4 \text{ cm} \)
\( h = 10 \text{ cm} \)

Common Mistakes

✕ Forgetting Units
Writing "84" instead of "\( 84 \text{ m}^2 \)." Always include squared or cubed units.
✕ Radius vs Diameter
Using diameter in \( \pi r^2 \) instead of radius. The radius is half the diameter.
✕ Forgetting \( \frac{1}{2} \)
In triangle area, skipping the \( \frac{1}{2} \) and just multiplying \( b \times h \). That gives twice the actual area.

Formula Reference Table

Shape Formula Variables
Rectangle Area \( A = l \times w \) \( l \) = length, \( w \) = width
Triangle Area \( A = \frac{1}{2} \times b \times h \) \( b \) = base, \( h \) = height
Circle Area \( A = \pi r^2 \) \( r \) = radius
Rect. Prism Vol. \( V = l \times w \times h \) \( l \), \( w \), \( h \)
Cylinder Vol. \( V = \pi r^2 h \) \( r \) = radius, \( h \) = height

Question 1 of 5

A rectangle has a length of 15 cm and a width of 8 cm. What is the area?

Question 2 of 5

A triangle has a base of 14 in and a height of 9 in. What is the area?

Question 3 of 5

A rectangular prism has \( l = 6 \), \( w = 4 \), and \( h = 5 \). What is the volume?

Question 4 of 5

A circle has a diameter of 10 cm. What is its area? Use \( \pi \approx 3.14 \).

Question 5 of 5

In the formula \( V = \pi r^2 h \), what does the variable \( h \) represent?