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Volume and Surface Area of Pyramids, Cones, Spheres

Learn how to find the volume and surface area of pyramids, cones, and spheres using the correct formulas. Let’s get started! 🚀

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

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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.

🛎️ Volume and Surface Area of a Pyramid

  • The volume of a pyramid is V=13BhV=\frac{1}{3}Bh, where BB is the base area and hh is the vertical height.
  • The surface area is A=B+LA=B+L, where LL is the total area of all triangular faces.

🛎️ Example: Pyramid Volume and Surface Area

  • With a square base of side 10 cm10\text{ cm} and height 12 cm12\text{ cm}, V=13(102)(12)=400 cm3V=\frac{1}{3}(10^2)(12)=400\text{ cm}^3.
  • Adding the base and four identical triangular faces gives a surface area of 360 cm2360\text{ cm}^2.

🛎️ Volume and Surface Area of a Cone

  • The volume of a cone is V=13πr2hV=\frac{1}{3}\pi r^2h, where rr is the radius and hh is the vertical height.
  • The surface area is A=πr2+πrsA=\pi r^2+\pi rs, where ss is the slant height.

🛎️ Example: Cone Volume and Surface Area

  • If r=3 cmr=3\text{ cm} and h=5 cmh=5\text{ cm}, then V=13π(32)(5)=15π cm3V=\frac{1}{3}\pi(3^2)(5)=15\pi\text{ cm}^3.
  • Using s=6 cms=6\text{ cm} gives a surface area of A=π(32)+π(3)(6)=27π cm2A=\pi(3^2)+\pi(3)(6)=27\pi\text{ cm}^2.

🛎️ Volume and Surface Area of a Sphere

  • The volume of a sphere is V=43πr3V=\frac{4}{3}\pi r^3, so the radius is cubed.
  • The surface area is A=4πr2A=4\pi r^2, so the radius is squared.
  • Exam tip: Volume uses cubed units and area uses squared units.

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