Cone and Pyramid Volume & Surface Area: Formulas and Examples

 

Cones and Pyramids: Volume and Surface Area Made Simple

Cones and pyramids share a useful volume shortcut, but their surface-area formulas need a little more care. This guide walks through both shapes, shows how radius, base area, vertical height, and slant height fit together, and gives you examples and practice to try.

The big idea: one-third of a matching solid

A cone has one-third the volume of a cylinder with the same circular base and vertical height. A pyramid has one-third the volume of a prism with the same base area and vertical height.

Cone volume: V = (1 ÷ 3) × π × r² × h

Pyramid volume: V = (1 ÷ 3) × B × h

On a small phone screen, swipe horizontally inside an equation box to see a long formula or calculation.

Know the measurements first

• Vertical height (h) is the perpendicular distance from the base to the tip or apex. Volume formulas use this height.
• Slant height (ℓ) follows a sloping side. It is used to calculate the cone’s curved surface and the triangular faces of a regular pyramid.
• For a right cone, r, h, and ℓ form a right triangle: ℓ = √(r² + h²).
• For a regular square pyramid, the face slant height comes from a right triangle with legs h and half the base side s: ℓ = √(h² + (s ÷ 2)²).

Surface-area formulas

The cone surface-area formulas below are for a right circular cone. For any pyramid, total surface area means its base plus every triangular side face; the perimeter shortcut applies to a regular pyramid.

Cone

The curved surface area leaves off the circular base. Total surface area includes that base.

Curved surface area = π × r × ℓ

Total surface area = π × r × ℓ + π × r²

Pyramid

Add the base area to the areas of every triangular side. For a regular pyramid, the triangular faces share one slant height, so you can use the perimeter shortcut.

Total surface area = B + sum of the triangular face areas

Regular pyramid: total surface area = B + (1 ÷ 2) × P × ℓ

Here, B is the base area, P is the base perimeter, and ℓ is the slant height measured along a face to the midpoint of a base edge.

A reliable step-by-step method

1. Identify the shape and the requested quantity. Decide whether the problem asks for volume, curved area, or total surface area.
2. Record the known measurements. For a cone, use its radius and vertical height for volume. If only diameter is given, divide it by 2 to get the radius.
3. Find any missing slant height with the right-triangle relationship when the shape is a right cone or a regular pyramid.
4. Substitute into the matching formula. Use the one-third formula for volume and the relevant outside faces for surface area.
5. Calculate, round at the end, and label units. Volume uses cubic units; surface area uses square units.

Worked example 1: right cone

A cone has radius 5 cm and vertical height 12 cm. Find its volume and total surface area. Use π ≈ 3.14.

Step 1: find slant height. The 5-12-13 right triangle gives ℓ = 13 cm.

ℓ = √(5² + 12²) = √169 = 13 cm

Volume

V = (1 ÷ 3) × 3.14 × 5² × 12 = 314 cm³

Total surface area

πrℓ + πr² = 3.14 × 5 × 13 + 3.14 × 5² = 204.1 + 78.5 = 282.6 cm²

The curved part alone is 204.1 cm²; including the bottom circle makes the total 282.6 cm².

Worked example 2: regular square pyramid

A regular square pyramid has base side 6 cm and vertical height 4 cm. Find its volume and total surface area.

Step 1: find base area and face slant height. The square base area is 6 × 6 = 36 cm². Half a base side is 3 cm, so the face slant height is 5 cm.

ℓ = √(4² + 3²) = √25 = 5 cm

Volume

V = (1 ÷ 3) × 36 × 4 = 48 cm³

Total surface area

B + (1 ÷ 2) × P × ℓ = 36 + (1 ÷ 2) × 24 × 5 = 96 cm²

The four triangular faces together have area 60 cm²; add the 36 cm² square base for 96 cm² total.

Memory tip: “one-third for volume”

When a cone or pyramid has the same base and vertical height as its matching cylinder or prism, its volume is one-third as large. Think “one-third for the pointy solid.” For surface area, do not use the one-third shortcut: add the actual outside faces.

Cone: V = (1 ÷ 3) × cylinder volume; pyramid: V = (1 ÷ 3) × matching prism volume

Common mistakes to avoid

• Putting slant height into the volume formula. Cone and pyramid volume use vertical height.
• Using diameter instead of radius. For a cone, halve the diameter before using πr².
• Leaving out the base in total surface area. Cone curved area and total area are different; pyramid total area includes its base.
• Using the regular-pyramid shortcut for an irregular pyramid. If side faces differ, calculate each triangular face and add them.
• Writing the wrong units. Use square units for surface area and cubic units for volume.
• Rounding too early. Keep precision until the final answer.

Practice questions

Use π ≈ 3.14. For each one, find volume and total surface area.

1. A right cone has radius 4 cm, vertical height 3 cm, and slant height 5 cm.
2. A regular square pyramid has base side 8 cm, vertical height 3 cm, and face slant height 5 cm.
3. A right cone has diameter 6 ft, vertical height 4 ft, and slant height 5 ft.

On a phone, swipe horizontally inside an equation box to see a long calculation.

Answers

1. Cone (r=4 cm, h=3 cm, ℓ=5 cm):
   V = (1 ÷ 3) × 3.14 × 4² × 3 = 50.24 cm³
   SA = 3.14 × 4 × 5 + 3.14 × 4² = 62.8 + 50.24 = 113.04 cm²
2. Square pyramid (side=8 cm, h=3 cm, ℓ=5 cm):
   V = (1 ÷ 3) × 64 × 3 = 64 cm³
   SA = 64 + (1 ÷ 2) × 32 × 5 = 144 cm²
3. Cone (diameter=6 ft, so r=3 ft; h=4 ft, ℓ=5 ft):
   V = (1 ÷ 3) × 3.14 × 3² × 4 = 37.68 ft³
   SA = 3.14 × 3 × 5 + 3.14 × 3² = 47.1 + 28.26 = 75.36 ft²

The takeaway

Use one-third × base area × vertical height for cone and pyramid volume. For surface area, add the curved or triangular sides—and include the base when total surface area is requested. Square units measure area; cubic units measure volume.