Introduction
- Pyramid and Cones are three-dimensional (3D) geometric shapes that play essential role in mathematics, architecture, engineering, and everyday life.
- Both shapes have a base and an apex, but they differ in structure and properties.
Real life Examples:
What is a Cone?
- A cone is a solid figure with a circular base that curves upward to meet at a single vertex, forming a pointed tip.
Key Features of a Cone:
- Base – A circular flat surface.
- Apex – The pointed top where all the sides meet.
- Height ($h$) – The perpendicular distance from the base to the apex.
- Slant Height ($\ell$) – The distance from the apex to any point on the edge of the base.
- Radius ($r$) – The distance from the center of the base to its edge.
What is a Pyramid?
- A Pyramid is a three-dimensional geometric shape with a polygonal base and triangular faces that meet at a common point called the apex.
Key Features of a Pyramid:
- Base – A polygon (e.g., triangle, square, pentagon, etc.).
- Apex – The topmost point where all triangular faces meet.
- Faces – Triangular sides connecting the base to the apex.
- Edges – The line segments where two faces meet.
- Height ($h$) – The perpendicular distance from the base to the apex.
- Slant Height ($\ell$) – The height of a triangular face from the base to the apex.
How to Find the Volume of Cones
- Volume is the amount of space occupied by an object.
Volume of a Cone:
- A cone has a circular base and tapers to apex.
- The formula for its volume is:
$$V = \frac{1}{3}\pi r^2 h$$
Where,
- $r$ = Radius of the base
- $h$ = Height (perpendicular distance from the base to the apex)
- $\pi \approx$ $3.1416$ (pi)
Steps to Calculate Volume:
1
Measure the radius ($r$) of the circular base.
2
Measure the height ($h$) of the cone.
3
Plug the values into the formula:
$$V = \frac{1}{3}\pi r^2 h$$
4
Compute the result.
How to Find the Volume of Pyramids
Volume of a Pyramid:
- A Pyramid has a polygonal base (e.g., square, triangle) and triangular faces that meet at an apex.
- The formula for its volume is:
$$V = \frac{1}{3}Bh$$
Where,
- $B$ = Area of the base
- $h$ = Height (perpendicular distance from the base to the apex)
Steps to Calculate Volume:
1
Find the area ($B$) of the base (depends on the base shape):
- Square base: $B = \text{side}^2$
- Rectangular base: $B = \text{length} \cdot \text{width}$
- Triangular base: $B = \frac{1}{2} \cdot \text{base} \cdot \text{height}$
2
Measure the height ($h$) of the pyramid.
3
Plug the values into the formula:
$$V = \frac{1}{3}Bh$$
4
Compute the result.
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Book a Consultation →How to Find the Surface Area of Cones
- Pyramid and Cones Surface area is the Total Area of all the surfaces (faces, bases, and curved sides) that cover a 3D object.
Surface area of a Cone:
It includes:
- Base area (a circle)
- Lateral surface area (the curved side)
Formula:
$$\text{Surface Area} = \text{Base Area} + \text{Lateral Area}$$
Where,
- Base Area = $\pi r^2$ (where $r = \text{radius}$)
- Lateral Area = $\pi r \ell$ (where $\ell = \text{slant height}$)
Steps to Calculate Surface Area:
1
Identify Given Values.
2
Find the Base Area.
3
Find the Lateral (Curved) Surface Area.
4
Calculate Total Surface Area.
How to Find the Surface Area of Pyramids
- Pyramid and Cones Surface area is the Total Area of all the surfaces (faces, bases, and curved sides) that cover a 3D object.
Surface Area of a Pyramid:
It includes:
- A base (which can be a square, triangle, rectangle, etc.)
- Triangular lateral faces (number depends on the base shape)
Formula:
$$\text{Surface Area} = \text{Base Area} + \text{Lateral Area}$$
Where,
- Lateral Area = $\frac{1}{2} \cdot \text{Perimeter of Base} \cdot \ell$
- $\ell$ = Slant height
Steps to Calculate Surface Area:
1
Identify Given Values.
2
Find the Base Area:
- Square base: $\text{Area} = s^2$ (side length $s$)
- Triangular base: $\text{Area} = \frac{1}{2}bh$ (base $b$, height $h$)
- Rectangular base: $\text{Area} = lw$ (length $l$, width $w$)
3
Find the lateral area.
4
Calculate Total Surface Area.
Solved Examples
Solved Example
Find the volume of a cone with radius 6 cm and height 10 cm. (Use $\pi \approx 3.14$)
SOLUTION
1
Given:
- $r = 6\text{ cm}$
- $h = 10\text{ cm}$
2
Plug the values into the formula:
$$V = \frac{1}{3}\pi r^2 h = \frac{1}{3} \cdot 3.14 \cdot (6)^2 \cdot 10$$
3
Compute the result:
$$V = \frac{1}{3} \cdot 3.14 \cdot 36 \cdot 10 = 376.8$$
Final Answer: $376.8\text{ cm}^3$
Solved Example
Find the volume of a pyramid with a square base of side length 9 meters and a height of 12 meters.
SOLUTION
1
Find the Area of the base:
- Since the base is a square:
- $\text{Base Area} = 9 \cdot 9 = 81$
2
Plug the values into the formula:
$$V = \frac{1}{3}Bh = \frac{1}{3} \cdot 81 \cdot 12$$
3
Compute the result:
$$V = 324$$
Final Answer: $324\text{ m}^3$
Solved Example
A cone has a radius of 5 cm and a slant height of 13 cm. Calculate its total surface area. Using $\pi \approx 3.14$
SOLUTION
1
Identify Given Values:
- Radius ($r$) = $5\text{ cm}$
- Slant height ($\ell$) = $13\text{ cm}$
2
Find the Base Area:
- The base is a circle, so its area is:
$$\text{Base Area} = \pi r^2 = \pi \cdot 5^2 = 25\pi$$
3
Find the Lateral (Curved) Surface Area:
- The lateral area of a cone is given by:
$$\text{Lateral Area} = \pi r \ell = \pi \cdot 5 \cdot 13 = 65\pi$$
4
Calculate Total Surface Area:
$$\text{Total Surface Area} = 25\pi + 65\pi = 90\pi$$
$$90 \cdot 3.14 = 282.6$$
Final Answer: $282.6\text{ cm}^2$
Solved Example
A square pyramid has a base side length of 6 m and a slant height of 5 m. Find its total surface area.
SOLUTION
1
Identify Given Values:
- Base side length ($s$) = $6\text{ m}$
- Slant height ($\ell$) = $5\text{ m}$
2
Find the Base Area:
- The base is a square, so its area is:
$$\text{Area} = s^2 = 6^2 = 36$$
3
Find the Lateral (Curved) Surface Area:
- A square pyramid has 4 triangular faces.
$$\text{Lateral Area} = \frac{1}{2} \cdot \text{Perimeter} \cdot \ell$$
Where,
$$\text{Perimeter} = 4 \cdot s = 4 \cdot 6 = 24$$
Now,
$$\text{Lateral Area} = \frac{1}{2} \cdot 24 \cdot 5 = 60$$
4
Calculate Total Surface Area:
$$\text{Total Surface Area} = 36 + 60 = 96$$
Final Answer: $96\text{ m}^2$
