## How to Find Volume and Surface Area of Pyramids and Cone

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### PYRAMIDS

A pyramid is a solid shape with a **polygonal** base and **triangular** sides that meet at a point called the "**apex**." The name of a pyramid comes from the shape of its ground. **Most** pyramids have a base that is a square or another **regular** polygon. This makes all sides of the pyramid the same shape, an isosceles triangle. The height is the **straight** line distance from the **top** to the middle of the bottom. The edge length and the **slant** height are other ways to measure pyramids. The edge length is the length of the sides of the triangles, and the slant height is the height of the **triangles**.

### Pyramid's volume

Generally, the volume of a pyramid with a base of area \(B\) and height \(h\) is: \(V \ = \ \frac{1}{3} \ B \ h\)

If the base is a **square** with sides that are each \(s\) long, then the volume is: \(V \ = \ \frac{1}{3} \ s^2 \ h\)

The volume of a pyramid is the same as that of a **prism** with the same **base** and **height**.

### Pyramid's Lateral Surface Area

If the base of a pyramid is a **regular** polygon with \(n\) sides, the length of each \(s\), and the slant height is 𝑙, then:

\(LSA \ = \ \frac{1}{2} \ n \ s \ l\)

If the square base, then: \(LSA \ = \ 2 \ s \ l\) \((n \ = \ 4)\)

### Pyramid's Total Surface Area

For the total area \((TSA)\), you simply put together **both** the base and lateral surface area:

\(TSA \ = \ LSA \ + \ B\)

If the **square** base, then: \(TSA \ = \ 2 \ s \ l \ + \ s^2\)

### CONES

Cone is a pyramid with a **circular** base.

You might be able to figure out a cone's **height** \(h\) (the distance from the top **perpendicular** to the base) or its **slant** height \(l\). (the length from the **apex** to the **edge** of the circular base). Note that the height, the radius, and the slant height form a right triangle, with the slant height as the hypotenuse. With the **Pythagorean** theorem, we can determine that the following things are the **same.**

The slant height \(l\), height \(h\), and radius \(r\) of a cone are related as follows:

- \(l \ = \ \sqrt{r^2 \ + \ h^2}\)
- \(h \ = \ \sqrt{l^2 \ - \ r^2}\)
- \(r \ = \ \sqrt{l^2 \ - \ h^2}\)

### Cone's volume

The volume of a cone is equal to one-third of the volume of a **cylinder** that has the **same** base and height. This is similar to the way that the volume of a pyramid is equal to one-third of the volume of a **prism** that has the **same** base and height.

A cone's volume is equal to one-third of the product of its **base** area with radius \(r\) and its height \(h\).

\(V \ = \ \frac{1}{3} \ πr^2 \ h\)

### Cone's Surface Area

\(LSA \ = \ πrl\)

\(TSA \ = \ LSA \ + \ πr^2 \ = \ πrl \ + \ πr^2\)

### Exercises for Pyramids and Cone

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## Pyramids and Cone Practice Quiz

### More Solid Figures courses

- How to Find the Volume of a Rectangular Prism
- How to Find the Volume of a Cylinder
- How to Find the Surface Area of a Cylinder
- How to Find the Surface Area of a Rectangular Prism
- How to Find Volume and Surface Area of Cubes
- How to Find the Volume of a Cube
- How to Find Volume and Surface Area of Pyramids and Cone