Volume and Surface Area of Pyramids, Cones, Spheres

Q: What is the difference between the height and the slant height?

The perpendicular height is measured straight from the apex to the base at a right angle. The slant height runs along a sloping face, from the base edge up to the apex. You use the perpendicular height for volume, and the slant height for surface area.

Q: Why can you use the same formula for the volume of a cone and a pyramid?

A cone works like a pyramid but with a circular base, so both use V = 1/3 × B × h. The only difference is the base area B: a pyramid uses its polygon base, while a cone uses π × r² for its circle.

Q: How do you find the surface area of a pyramid?

Add the base area to the areas of all the triangular faces. For a square pyramid, work out the square base area, find the area of one triangular face using its slant height, multiply that by four, then add the base.

Q: Do you use r squared or r cubed for a sphere?

Use r cubed for the volume, V = 4/3 × π × r³, and r squared for the surface area, A = 4 × π × r². Swapping the two is the most common slip, so match the power to what you want: r cubed for volume, r squared for surface area.

Q: What is the curved surface area of a cone?

The curved surface is the cone's lateral surface, the part that wraps from the base up to the apex. Unfolded, it forms a sector of a circle, and its area is π × r × s, where s is the slant height. Add the circular base for the total surface area.

Key concept

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.

Volume and Surface Area of Pyramids, Cones, Spheres - introduction visual

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Flashcards

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Diagram showing volume and surface area formulas for a pyramid. Includes base area, height, and the net of the pyramid.

Illustrating the volume and surface area of a square-based pyramid, including calculations for volume and surface area along with the pyramid's net.

Square pyramid with base 10 cm, height 12 cm, slant 13 cm; net shown; volume 400 cm³, surface area 360 cm².

The volume and surface area of a cone, showing formulas for volume (V = 1/3 × B × h) and surface area (A = B + L), with net representation.

Cone with radius 3 cm, height 5 cm, slant height 6 cm. Volume is calculated to be 15π cm³ and surface area is 27π cm².

Cross-section of a sphere with a radius of 3 cm, showing formulas for volume and surface area.

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.

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What is the formula for the volume of a pyramid?

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Interactive Activity

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base side (s)

height (h)

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Students Also Ask

The questions students bump into most on this topic

What is the difference between the height and the slant height?

The perpendicular height is measured straight from the apex to the base at a right angle. The slant height runs along a sloping face, from the base edge up to the apex. You use the perpendicular height for volume, and the slant height for surface area.

Why can you use the same formula for the volume of a cone and a pyramid?

A cone works like a pyramid but with a circular base, so both use V = 1/3 × B × h. The only difference is the base area B: a pyramid uses its polygon base, while a cone uses π × r² for its circle.

How do you find the surface area of a pyramid?

Add the base area to the areas of all the triangular faces. For a square pyramid, work out the square base area, find the area of one triangular face using its slant height, multiply that by four, then add the base.

Do you use r squared or r cubed for a sphere?

Use r cubed for the volume, V = 4/3 × π × r³, and r squared for the surface area, A = 4 × π × r². Swapping the two is the most common slip, so match the power to what you want: r cubed for volume, r squared for surface area.

What is the curved surface area of a cone?

The curved surface is the cone's lateral surface, the part that wraps from the base up to the apex. Unfolded, it forms a sector of a circle, and its area is π × r × s, where s is the slant height. Add the circular base for the total surface area.

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