Surface Area of a Pyramid & Volume of Square Pyramids & Triangular Pyramids

Summary

This video tutorial covers how to calculate the volume, surface area, and lateral area of square-based and triangular pyramids. It explains the key formulas, such as the volume formula for a square-based pyramid (V = 1/3 × b² × h) and the surface area formula (A = b² + 2bl). The video also provides step-by-step instructions on finding the slant height using the Pythagorean theorem. Additionally, the tutorial demonstrates how to apply these concepts in examples, including a triangular pyramid, to reinforce the learning process. Clear explanations and practical examples make complex geometry topics accessible.

Takeaways

• 😀 The volume of a square-based pyramid is calculated using the formula: V = (1/3) × b² × h, where 'b' is the side length of the base and 'h' is the height of the pyramid.
• 😀 To calculate the surface area of a square-based pyramid, use the formula: S = b² + 2bl, where 'b' is the side length of the base and 'l' is the slant height.
• 😀 The volume of a triangular pyramid is given by the formula: V = (1/3) × Area of base × h, where the base is a triangle and 'h' is the height of the pyramid.
• 😀 The lateral area of a square-based pyramid consists of four triangles, and the formula for lateral area is: 2 × b × l, where 'b' is the base side and 'l' is the slant height.
• 😀 The slant height ('l') is different from the height ('h') of the pyramid. The slant height is used in surface area calculations, while the height is used for volume.
• 😀 To find the slant height of a square-based pyramid, you can use the Pythagorean theorem if you know the height and half the base side.
• 😀 When calculating the volume of a pyramid, the base area is crucial. For a square-based pyramid, the base area is b², and for a triangular pyramid, the base area is 1/2 × base × height of the triangle.
• 😀 For surface area, the area of the base is added to the lateral area, which involves calculating the area of each triangular face and multiplying it by 4 for a square-based pyramid.
• 😀 In triangular pyramids, particularly those with an equilateral triangle as the base, the formula for the area of the base is (√3/4) × b².
• 😀 Be cautious with the terminology: 'base' refers to the bottom face of the pyramid, while 'height' refers to the perpendicular distance from the apex to the base, and 'slant height' refers to the length along the face of the pyramid.

Q & A

What is the formula for finding the volume of a square-based pyramid?

The formula for the volume of a square-based pyramid is V = 1/3 × b² × h, where 'b' is the side length of the base and 'h' is the height of the pyramid.

How do you calculate the surface area of a square-based pyramid?

The surface area of a square-based pyramid is calculated using the formula A = b² + 2bl, where 'b' is the side length of the base and 'l' is the slant height of the pyramid.

What is the difference between the height and the slant height of a pyramid?

The height of a pyramid is the perpendicular distance from the apex to the center of the base, while the slant height is the distance from the apex to a point on the edge of the base along the triangular face.

How do you find the lateral area of a square-based pyramid?

The lateral area of a square-based pyramid is found using the formula 2bl, where 'b' is the side length of the base and 'l' is the slant height.

How is the volume of a triangular pyramid calculated?

The volume of a triangular pyramid is calculated using the formula V = 1/3 × (1/2 × base × height of the triangle) × h, where the base and height are the dimensions of the triangular base and 'h' is the height of the pyramid.

Why is the formula for the volume of a pyramid multiplied by 1/3?

The factor of 1/3 comes from the relationship between the volume of a pyramid and a prism. The volume of a pyramid is one-third the volume of a prism with the same base area and height.

What is the formula for the surface area of a triangular pyramid with an equilateral triangle base?

The surface area of a triangular pyramid with an equilateral triangle base is A = (√3/4) × b² + 3/2 × b × l, where 'b' is the side length of the equilateral triangle and 'l' is the slant height.

What steps are needed to find the slant height of a square-based pyramid?

To find the slant height of a square-based pyramid, you can use the Pythagorean theorem. First, divide the base length 'b' by 2 to get half the side of the base, then use the height 'h' and half the base length to calculate the slant height using l = √((b/2)² + h²).

In the example with a square base pyramid, if the base side length is 6 units and the height is 10 units, what is the volume?

Using the formula for volume V = 1/3 × b² × h, with b = 6 and h = 10, the volume is V = 1/3 × 6² × 10 = 120 cubic units.

What is the lateral area of a square-based pyramid with a base of 8 cm and a slant height of 15 cm?

The lateral area is calculated using the formula 2bl. With b = 8 cm and l = 15 cm, the lateral area is 2 × 8 × 15 = 240 square centimeters.