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Solve Area and
Volume Problems

Master the formulas for area, volume, and surface area of 2D and 3D shapes: triangles, quadrilaterals, polygons, cubes, and right prisms.

7th Grade Math

What Is Area?

Area measures the amount of flat space inside a two-dimensional shape. We express it in square units such as \(\text{cm}^2\), \(\text{m}^2\), or \(\text{ft}^2\).

\(9\) units²
Square: \( A = s^2 \)
\(h\)
\(b\)
Triangle: \( A = \tfrac{1}{2}bh \)

What Is Volume?

Volume measures the amount of space inside a three-dimensional object. We express it in cubic units such as \(\text{cm}^3\), \(\text{m}^3\), or \(\text{ft}^3\).

\(l\)
\(h\)
\(w\)
Cube / Rectangular Prism
Triangular Prism

What Is Surface Area?

Surface area is the total area of all the faces (outer surfaces) of a 3D object. Imagine unfolding a box flat: the surface area is the area of that entire flat shape (the net).

bottom
top
front
left
right
back
Net of a rectangular prism: add the area of every face

Core Formulas: 2D Shapes

Rectangle
\( A = l \times w \)
Triangle
\( A = \tfrac{1}{2} b h \)
Parallelogram
\( A = b \times h \)
Trapezoid
\( A = \tfrac{1}{2}(b_1 + b_2) h \)

Core Formulas: 3D Shapes

Cube
\( V = s^3 \)
\( SA = 6s^2 \)
Rectangular Prism
\( V = lwh \)
\( SA = 2(lw + lh + wh) \)
Triangular Prism
\( V = \tfrac{1}{2}bh \cdot l \)
\( SA = bh + \text{(3 rect. faces)} \)

Area vs. Volume vs. Surface Area

Area Volume Surface Area
Measures Flat space inside Space inside a 3D object Total outer covering
Dimensions 2D 3D 3D (sum of 2D faces)
Units \(\text{units}^2\) \(\text{units}^3\) \(\text{units}^2\)
Real-world example Paint a wall Fill a box Wrap a gift

How to Solve These Problems

1
Identify the shape. Is it 2D (flat) or 3D (solid)? Name it: triangle, rectangle, prism, cube, etc.
2
Decide what is being asked. Area, volume, or surface area? Check the units in the answer choices.
3
Write the formula. Choose the correct formula for the shape and measurement type.
4
Substitute the values. Plug in the given numbers for base, height, length, width, etc.
5
Compute and label. Calculate the answer and attach the correct units (\(\text{cm}^2\) or \(\text{cm}^3\)).

Example 1: Triangle Area

Find the area of a triangle with a base of 10 cm and a height of 6 cm.

10 cm
6 cm

Example 2: Rectangular Prism Volume

A storage box is 8 in long, 5 in wide, and 3 in tall. What is its volume?

8 in
3 in
5 in

Example 3: Surface Area

Using the same box from Example 2 (\( l = \) 8, \( w = \) 5, \( h = \) 3), find its surface area.

Example 4: Triangular Prism Volume

A tent has a triangular cross-section with base 6 ft and height 4 ft. The tent is 10 ft long. Find the volume.

6 ft
4 ft
10 ft

Composite Shapes

Real-world problems often involve composite shapes: shapes made by combining two or more basic shapes. To find the total area, decompose the figure and add (or subtract) the parts.

6 m
9 m
5 m
5 m
Split into
Rectangle A
\(6 \times 9 = 54\) m²
Rectangle B
\(5 \times 5 = 25\) m²
Total = 79 m²

Sometimes You Subtract

If a shape has a hole or a piece removed, find the area of the whole shape first, then subtract the area of the removed part.

10 m
8 m
\(4 \times 4\) m
Whole: \(10 \times 8 = 80\) m²
Cutout: \(4 \times 4 = 16\) m²
Shaded = \(80 - 16\)
= 64 m²

Common Mistakes

✕ Wrong Height
Using the slant side of a triangle instead of the perpendicular height. The height must form a 90° angle with the base.
✕ Unit Mix-Up
Writing \(\text{cm}^2\) when the answer is a volume (\(\text{cm}^3\)), or vice versa. Always check: is it area or volume?
✕ Forgetting \(\tfrac{1}{2}\)
Calculating \( b \times h \) for a triangle and forgetting to multiply by \(\tfrac{1}{2}\). That gives the parallelogram area, not the triangle's.

Formula Summary

Shape Area / Volume Surface Area
Rectangle \( A = lw \) —
Triangle \( A = \tfrac{1}{2}bh \) —
Trapezoid \( A = \tfrac{1}{2}(b_1+b_2)h \) —
Cube \( V = s^3 \) \( SA = 6s^2 \)
Rect. Prism \( V = lwh \) \( SA = 2(lw+lh+wh) \)
Tri. Prism \( V = \tfrac{1}{2}bh \cdot l \) Sum of all faces

Question 1 of 5

A triangle has a base of 12 cm and a height of 9 cm. What is its area?

Question 2 of 5

A rectangular prism has \( l = 7 \text{ m} \), \( w = 4 \text{ m} \), \( h = 3 \text{ m} \). What is its volume?

Question 3 of 5

A cube has a side length of 5 cm. What is its surface area?

Question 4 of 5

A triangular prism has a triangular base with \( b = 6 \text{ in} \) and \( h = 4 \text{ in} \). The prism is 15 in long. What is its volume?

Question 5 of 5

A rectangular garden is 12 m by 8 m. A square fountain (\( 2 \text{ m} \times 2 \text{ m} \)) sits in the middle. What is the area of the remaining garden?