Geometry and Measures

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Properties of Circles

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Constructing Triangles

Congruent Triangles

Thales’ Theorem

Perpendicular Bisectors and Circumcircle

Angle Bisectors and Incircle

Volume and Surface Area

Solids and Units of Volume

Volume of a Cuboid and Cube

Surface Area of Solids

Volume of Prisms and Cylinders

Cavalieri’s Principle

Volume and Surface Area of Pyramids, Cones, Spheres

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Volume and Surface Area of Pyramids, Cones, Spheres

Geometry and Measures Volume and Surface Area of Pyramids, Cones, Spheres

Courses Overview

🎬 Video Tutorial

(0:18) Volume of Pyramids and Cones: $\frac{1}{3} \times \text{base area} \times \text{height}$, where height is always perpendicular to the base.
(0:37) Surface Area of Pyramids: Add the base area and the areas of all triangular sides.
(1:50) Surface Area of Cones: Add the base circle area $(\pi \times r^2)$ and the lateral surface area ($\pi \times r \times s$), where $s$ is the slant height.
(3:00) Volume and Surface Area of Spheres: Remember, volume requires $r^3$, while surface area needs $r^2$.

📂 Revision Cards

Diagram showing volume and surface area formulas for a pyramid. Includes base area, height, and the net of the pyramid. — 1) Volume and Surface Area of Pyramids (Formula)

Illustrating the volume and surface area of a square-based pyramid, including calculations for volume and surface area along with the pyramid's net. — 2) Volume and Surface Area of Pyramids (Example)

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. — 3) Volume and Surface Area of a Cone (Formula)

A cone with a radius of 3 cm, height of 5 cm, and slant height of 6 cm. The volume is calculated as fifteen pi cm³, and the surface area as twenty-seven pi cm². — 4) Volume and Surface Area of a Cone (Example)

Cross-section of a sphere with a radius of 3 cm, showing formulas for volume and surface area. — 5) Volume and Surface Area of Spheres (Formula & Example)

🍪 Quiz Time - Practice Now!

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Volume and Surface Area of Pyramids, Cones, Spheres

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Q: What is the formula for the volume of a pyramid?

$V = 4\pi r^2$

$V = \frac{1}{3} B \times h$

$V = \frac{1}{3} \pi r^2 h$

$V = B \times h$

Feedback: Not quite! The formula for the volume of a pyramid is $V = \frac{1}{3} B \times h$, where $B$ is the area of the base and $h$ is the height of the pyramid.

Feedback: Well done! The formula for the volume of a pyramid is $V = \frac{1}{3} B \times h$, where $B$ is the area of the base and $h$ is the height of the pyramid.

2 / 6

Q: A pyramid has a triangular base with an area of $30 \, \text{cm}^2$ and a height of 10 cm. What is its volume?

$150 \, \text{cm}^3$

$100 \, \text{cm}^3$

$75 \, \text{cm}^3$

$50 \, \text{cm}^3$

Feedback: Not quite! The volume of a pyramid is $V = \frac{1}{3} B \times h$, where $B$ is the base area and $h$ is the height. Substituting the values, $V = \frac{1}{3} \times 30 \times 10 = 100 \, \text{cm}^3$.

Feedback: Well done! The volume of a pyramid is $V = \frac{1}{3} B \times h$, where $B$ is the base area and $h$ is the height. Substituting the values, $V = \frac{1}{3} \times 30 \times 10 = 100 \, \text{cm}^3$.

3 / 6

Q: What is the surface area of a square pyramid with a base area of $16 \, \text{cm}^2$ and four identical triangle faces with an area of $10 \, \text{cm}^2$ each?

$48 \, \text{cm}^2$

$72 \, \text{cm}^2$

$56 \, \text{cm}^2$

$64 \, \text{cm}^2$

Feedback: Not quite! To calculate the surface area of the square pyramid, add the area of the base to the area of the four triangular faces: $16 \, \text{cm}^2 + 4 \times 10 \, \text{cm}^2 = 56 \, \text{cm}^2$.

Feedback: Well done! To calculate the surface area of the square pyramid, add the area of the base to the area of the four triangular faces: $16 \, \text{cm}^2 + 4 \times 10 \, \text{cm}^2 = 56 \, \text{cm}^2$.

4 / 6

Q: A sphere has a radius of 6 cm. What is its surface area?

$36\pi \, \text{cm}^2$

$72\pi \, \text{cm}^2$

$216\pi \, \text{cm}^2$

$144\pi \, \text{cm}^2$

Feedback: Not quite! The formula for the surface area of a sphere is $A = 4\pi r^2$. Substituting the radius, $A = 4\pi (6)^2 = 144\pi \, \text{cm}^2$.

Feedback: Well done! The formula for the surface area of a sphere is $A = 4\pi r^2$. Substituting the radius, $A = 4\pi (6)^2 = 144\pi \, \text{cm}^2$.

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Q: What is the volume of a cone with a radius of 3 cm and a height of 5 cm?

$30\pi \, \text{cm}^3$

$45\pi \, \text{cm}^3$

$15\pi \, \text{cm}^3$

$75\pi \, \text{cm}^3$

Feedback: Not quite! The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$. Substituting the values, $V = \frac{1}{3} \pi (3)^2 (5) = 15\pi \, \text{cm}^3$.

Feedback: Well done! The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$. Substituting the values, $V = \frac{1}{3} \pi (3)^2 (5) = 15\pi \, \text{cm}^3$.

6 / 6

Q: A cone has a slant height of 8 cm and a radius of 4 cm. What is the surface area?

$32\pi \, \text{cm}^2$

$48\pi \, \text{cm}^2$

$64\pi \, \text{cm}^2$

$80\pi \, \text{cm}^2$

Feedback: Not quite! The formula for the surface area of a cone is $A = \pi r^2 + \pi r s$. This formula comes from adding the area of the base (a circle, which is $\pi r^2$) and the lateral surface area (which unfolds into a sector of a circle and has an area of $\pi r s$). Substituting the values, we get $A = \pi (4)^2 + \pi (4)(8) = 16\pi + 32\pi = 48\pi \, \text{cm}^2$.

Feedback: Well done! The formula for the surface area of a cone is $A = \pi r^2 + \pi r s$. This formula comes from adding the area of the base (a circle, which is $\pi r^2$) and the lateral surface area (which unfolds into a sector of a circle and has an area of $\pi r s$). Substituting the values, we get $A = \pi (4)^2 + \pi (4)(8) = 16\pi + 32\pi = 48\pi \, \text{cm}^2$.

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