Volume and Surface Area of Pyramids, Cones, Spheres
The volume of a pyramid is V = ⅓ × base area × height, the same formula a cone uses. A base of 30 cm² and height 10 cm gives 100 cm³. Its surface area is A = B + L, the base area plus all its triangular faces.
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What Is a Pyramid?
- A pyramid is a 3D shape with a polygon base and triangular faces meeting at a single point.
- The base can be a triangle, square, or any other polygon.
Volume and Surface Area of a Pyramid
- The volume of a pyramid is , where B is the base area and h is the perpendicular height.
- The surface area is , where L is the total area of all triangular faces.
Example: Pyramid Volume and Surface Area
- With a square base of side and height , V .
- Adding the base and four identical triangular faces gives a surface area of .
Volume and Surface Area of a Cone
- The volume of a cone is , where r is the radius and h is the perpendicular height.
- The surface area is , where s is the slant height.
Example: Cone Volume and Surface Area
- If cm and cm, then .
- Using cm gives a surface area of .
Volume and Surface Area of a Sphere
- The volume of a sphere is , so the radius is cubed.
- The surface area is , so the radius is squared.
- Exam tip: Volume uses cubed units and area uses squared units.
Practice Questions
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What is the formula for the volume of a pyramid?
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The volume of a pyramid is , where B is the area of the base (whatever shape the base is) and h is the perpendicular height from the base up to the apex.
A pyramid fills exactly one third of the prism that has the same base and the same height, which is where the comes from.
A pyramid has a triangular base with an area of and a height of . What is its volume?
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The volume of a pyramid is , where B is the base area and h is the perpendicular height.
Substituting: .
What is the surface area of a square pyramid with a base area of and four identical triangular faces with an area of each?
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The surface area is the square base plus all four triangular faces, including the two you cannot see in the drawing.
. A pyramid has only one base, so the is counted once.
A sphere has a radius of . What is its surface area?
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The surface area of a sphere is . Square the radius first: .
(about ).
What is the volume of a cone with a radius of and a height of ?
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The volume of a cone is . Square the radius first: .
(about ).
A cone has a slant height of and a radius of . What is its surface area?
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The surface area of a cone is the base circle plus the curved surface: , where l is the slant height.
Base: . Curved surface: . Total: (about ).
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Frequently asked questions
The questions students bump into most on this topic
Volume = ⅓ × base area × perpendicular height, whatever shape the base is. A square pyramid with a 10 cm side and a height of 12 cm has a base area of 100 cm², so V = ⅓ × 100 × 12 = 400 cm³. A pyramid on a triangular base of 30 cm² with a height of 10 cm has V = ⅓ × 30 × 10 = 100 cm³. Use the height, not the slant edge.
A cone is a pyramid with a circular base, so it uses the same formula: V = ⅓ × base area × height, with the base area πr². For a cone with radius 3 cm and height 5 cm, V = ⅓ × π × 9 × 5 = 15π cm³. The third is there for the same reason as in a pyramid: a pointed solid holds a third of the prism on the same base.
Add the base to all the triangular faces. For the square pyramid with a 10 cm side, the base is 100 cm². Each face is a triangle with base 10 cm and height 13 cm, the slant height, so its area is ½ × 10 × 13 = 65 cm²; four faces make 260 cm². The surface area is 100 + 260 = 360 cm².
Add the circular base, πr², to the curved surface, πrs, where s is the slant height: A = πr² + πrs. For a cone with r = 3 cm and s = 6 cm, A = 9π + 18π = 27π cm². The curved part unfolds into a sector of a circle, which is where πrs comes from.
Volume = 4/3 × π × r³ and surface area = 4 × π × r². For a sphere with radius 3 cm, V = 4/3 × π × 27 = 36π cm³ and A = 4 × π × 9 = 36π cm². The common slip is swapping the powers: cubed for volume, squared for area. A sphere of radius 6 cm has surface area 4 × π × 36 = 144π cm².
The perpendicular height goes straight from the apex down to the base at a right angle; the slant height runs along a sloping face from the base edge to the apex. Use the height for volume and the slant height for surface area: the square pyramid has height 12 cm but slant height 13 cm. If only one is given, Pythagoras links them.
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