Surface Area of 3D Shapes — Prisms, Cylinders, and More

Grade: 8-9 | Topic: Geometry

What You Will Learn

After this lesson you will be able to calculate the total surface area of the most common 3D shapes: rectangular prisms (boxes), triangular prisms, cylinders, pyramids, cones, and spheres. You will understand the difference between lateral area and total surface area, and apply these formulas to real-world wrapping, painting, and construction problems.

Theory

What is surface area?

The surface area (SA) of a 3D shape is the total area of all the surfaces that form the outside of the shape. Imagine unfolding or "unwrapping" the shape into a flat pattern (called a net) — the surface area is the combined area of every piece in that net.

Surface area is measured in square units (cm$^2$, m$^2$, ft$^2$) because it measures area, not volume.

Total vs. lateral surface area

• Total surface area includes every face — top, bottom, and sides.
• Lateral surface area includes only the sides (no top or bottom bases).

Many problems ask specifically for one or the other, so always read carefully.

Rectangular prism (box)

A rectangular prism has three pairs of identical rectangular faces. If the length is $l$, width is $w$, and height is $h$:

$\text{SA} = 2lw + 2lh + 2wh$

This accounts for the top/bottom ($2lw$), front/back ($2lh$), and left/right sides ($2wh$).

Lateral area (sides only, no top/bottom):

$\text{LA} = 2lh + 2wh = 2h(l + w)$

For example, a box that is 5 cm by 4 cm by 3 cm:

$\text{SA} = 2(5)(4) + 2(5)(3) + 2(4)(3) = 40 + 30 + 24 = 94 \text{ cm}^2$

Cube

A cube is a special rectangular prism where $l = w = h = s$:

$\text{SA} = 6s^2$

Triangular prism

A triangular prism has two identical triangular bases and three rectangular side faces. If the triangular base has sides $a$, $b$, $c$ and area $A_{\text{base}}$, and the prism has length (depth) $L$:

$\text{SA} = 2A_{\text{base}} + (a + b + c) \times L$

The term $(a + b + c) \times L$ is the lateral area — the perimeter of the base multiplied by the length.

Cylinder

A cylinder has two circular bases and a curved lateral surface. If you unroll the curved surface, it becomes a rectangle with width $2\pi r$ (the circumference) and height $h$:

$\text{SA} = 2\pi r^2 + 2\pi rh$

• $2\pi r^2$ = area of the two circular bases
• $2\pi rh$ = lateral (curved) surface area

For example, a cylinder with radius 3 cm and height 10 cm:

$\text{SA} = 2\pi(3)^2 + 2\pi(3)(10) = 18\pi + 60\pi = 78\pi \approx 245.04 \text{ cm}^2$

Pyramid (square base)

A pyramid with a square base of side $s$ and a slant height $l$ (the distance from the base edge to the apex along the face):

$\text{SA} = s^2 + 2sl$

• $s^2$ = base area
• $2sl$ = lateral area (four triangular faces, each with area $\frac{1}{2} \times s \times l$, totaling $4 \times \frac{1}{2}sl = 2sl$)

Cone

A cone has a circular base and a curved lateral surface:

$\text{SA} = \pi r^2 + \pi r l$

where $r$ is the radius and $l$ is the slant height. If you are given the vertical height $h$ instead, find the slant height using the Pythagorean theorem:

$l = \sqrt{r^2 + h^2}$

Sphere

A sphere has no edges or bases — just one curved surface:

$\text{SA} = 4\pi r^2$

This formula says the surface area of a sphere is exactly four times the area of a circle with the same radius.

Worked Examples

Example 1: Surface area of a rectangular prism (easy)

Problem: A box is 8 cm long, 6 cm wide, and 4 cm tall. Find its total surface area.

Step 1: Identify the dimensions. $l = 8$, $w = 6$, $h = 4$.

Step 2: Apply the formula. $\text{SA} = 2lw + 2lh + 2wh = 2(8)(6) + 2(8)(4) + 2(6)(4)$

Step 3: Calculate each term. $= 96 + 64 + 48 = 208 \text{ cm}^2$

Answer: 208 cm$^2$

Example 2: Surface area of a cylinder (easy)

Problem: A tin can has a radius of 5 cm and a height of 12 cm. Find its total surface area. Use $\pi \approx 3.14$.

Step 1: Calculate the base areas. $2\pi r^2 = 2 \times 3.14 \times 25 = 157 \text{ cm}^2$

Step 2: Calculate the lateral area. $2\pi rh = 2 \times 3.14 \times 5 \times 12 = 376.8 \text{ cm}^2$

Step 3: Add them together. $\text{SA} = 157 + 376.8 = 533.8 \text{ cm}^2$

Answer: 533.8 cm$^2$

Example 3: Surface area of a triangular prism (medium)

Problem: A triangular prism has a right-triangular base with legs 3 cm and 4 cm (hypotenuse 5 cm). The prism is 10 cm long. Find the total surface area.

Step 1: Find the area of the triangular base. $A_{\text{base}} = \frac{1}{2} \times 3 \times 4 = 6 \text{ cm}^2$

Step 2: Find the lateral area using the base perimeter. $\text{Perimeter} = 3 + 4 + 5 = 12 \text{ cm}$ $\text{LA} = 12 \times 10 = 120 \text{ cm}^2$

Step 3: Calculate total surface area. $\text{SA} = 2(6) + 120 = 12 + 120 = 132 \text{ cm}^2$

Answer: 132 cm$^2$

Example 4: Surface area of a cone (medium)

Problem: A cone has a radius of 6 cm and a vertical height of 8 cm. Find its total surface area. Use $\pi \approx 3.14$.

Step 1: Find the slant height using the Pythagorean theorem. $l = \sqrt{r^2 + h^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}$

Step 2: Calculate the base area. $\pi r^2 = 3.14 \times 36 = 113.04 \text{ cm}^2$

Step 3: Calculate the lateral area. $\pi r l = 3.14 \times 6 \times 10 = 188.4 \text{ cm}^2$

Step 4: Add them together. $\text{SA} = 113.04 + 188.4 = 301.44 \text{ cm}^2$

Answer: 301.44 cm$^2$

Example 5: Surface area of a sphere — painting a dome (challenging)

Problem: A hemispherical dome has a diameter of 14 m. A painter needs to paint the curved outer surface (not the flat base). How many square metres must be painted? If one litre of paint covers 10 m$^2$, how many litres are needed? Use $\pi \approx 3.14$.

Step 1: Find the radius. $r = \frac{14}{2} = 7 \text{ m}$

Step 2: The curved surface area of a hemisphere is half the surface area of a full sphere. $A_{\text{curved}} = \frac{1}{2} \times 4\pi r^2 = 2\pi r^2 = 2 \times 3.14 \times 49 = 307.72 \text{ m}^2$

Step 3: Calculate litres of paint needed. $\text{Litres} = \frac{307.72}{10} = 30.772 \approx 31 \text{ litres (round up)}$

Answer: Curved area = 307.72 m$^2$, Paint needed = 31 litres

Common Mistakes

Mistake 1: Confusing surface area with volume

❌ "Surface area of a box = $l \times w \times h$" (this is volume)

✅ Surface area of a box = $2lw + 2lh + 2wh$

Why this matters: Volume measures three-dimensional space (in cubic units). Surface area measures the two-dimensional skin of the object (in square units). Multiplying all three dimensions gives volume, not surface area.

Mistake 2: Forgetting to include all faces

❌ For a cylinder, only calculating the lateral area: $\text{SA} = 2\pi rh$

✅ Total SA includes the two circular bases too: $\text{SA} = 2\pi r^2 + 2\pi rh$

Why this matters: A closed cylinder has three surfaces — the curved wall and two circular lids. Forgetting the bases underestimates the surface area. (Note: if the problem says "open-top cylinder," then you subtract one base.)

Mistake 3: Using vertical height instead of slant height for cones and pyramids

❌ Cone lateral area with vertical height: $\pi r h = 3.14 \times 6 \times 8 = 150.72$

✅ Use slant height: $l = \sqrt{6^2 + 8^2} = 10$. Lateral area: $\pi r l = 3.14 \times 6 \times 10 = 188.4$

Why this matters: The lateral surface of a cone unfolds into a sector of a circle. The radius of that sector is the slant height, not the vertical height. Using the wrong height gives a smaller (incorrect) answer.

Practice Problems

Try these on your own before checking the answers:

1. A rectangular prism is 10 cm by 7 cm by 5 cm. Find its total surface area.
2. A cube has a side length of 9 m. Find its surface area.
3. A cylinder has a radius of 4 cm and a height of 15 cm. Find the total surface area. (Use $\pi \approx 3.14$.)
4. A cone has a radius of 3 cm and a slant height of 7 cm. Find the total surface area. (Use $\pi \approx 3.14$.)
5. A basketball has a diameter of 24 cm. Find its surface area. (Use $\pi \approx 3.14$.)

Click to see answers

1. $\text{SA} = 2(10)(7) + 2(10)(5) + 2(7)(5) = 140 + 100 + 70 = 310$ cm$^2$.
2. $\text{SA} = 6(9)^2 = 6 \times 81 = 486$ m$^2$.
3. $\text{SA} = 2(3.14)(16) + 2(3.14)(4)(15) = 100.48 + 376.8 = 477.28$ cm$^2$.
4. Base: $3.14 \times 9 = 28.26$ cm$^2$. Lateral: $3.14 \times 3 \times 7 = 65.94$ cm$^2$. Total: $28.26 + 65.94 = 94.2$ cm$^2$.
5. $r = 12$ cm. $\text{SA} = 4 \times 3.14 \times 144 = 1808.64$ cm$^2$.

Summary

• Surface area is the total area of all outer faces of a 3D shape, measured in square units.
• Rectangular prism: $\text{SA} = 2lw + 2lh + 2wh$. Cube: $\text{SA} = 6s^2$.
• Cylinder: $\text{SA} = 2\pi r^2 + 2\pi rh$ (two bases + curved wall).
• Cone: $\text{SA} = \pi r^2 + \pi rl$ (use slant height, not vertical height).
• Sphere: $\text{SA} = 4\pi r^2$.
• Always check whether a problem asks for total surface area (all faces) or lateral surface area (sides only).

Related Topics

• Area and Perimeter — Formulas, Examples, and Practice
• Area and Circumference of a Circle — Formulas and Examples
• Pythagorean Theorem — Formula, Proof, and Examples

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