AREA & VOLUME OF CIRCULAR SOLIDS |

**Circular Solids****: Cylinders, Cones and Spheres**

These solids are based on circles. Cans, the columns on our buildings, storage tanks for liquids are all **cylindrical** shapes. Highway cones and dixie cups are **conical** shapes, and our planet and the balls we play with (except footballs) are **spherical** shapes.

**Circle Formulas:**(*r* = radius, *d* = diameter**.**)

Perimeter or ** circumference** of a circle: ,

** Area** of a circle: ,

**Cylinders**:

A ** cylinder**, is a

Another way to write the formula for Area of a Cylinder is:

**Cones**:

A ** cone**'s base is a circular flat surface. The "sides" (curved surface) come to a

**The surface area** of a cone is the **area of the circular base plus the area of the circle sector** that makes up the sides. The **area of** circle's **sector depends on** the measure of **the vertex angle**.

We find the area of a sector created by a central angle of a circle with this proportion

.

The volume of a frustrum is the volume of the entire cone minus the volume of the cone that's been cut off the top.

**Spheres:**

**the height and radius of a sphere are equal so the only measure we need to find both Area and Volume is the radius.**

A ** sphere or orb** is the correct term for a ball or globe.

It has no flat surfaces.

Northern Hemisphere -- the upper half of the sphere on which we all live.

The **curved surface** of a sphere is made up of **4 circles**. The lacing

on a softball almost outlines the **4 circular areas** that make up the **surface**.

From this image, we can see how the sphere is made of 4 identical cones with bases on the

4 circular surfaces, with vertices meeting at the center of the sphere.

**Decomposable Solids:**

Find the area or volume of the separate parts then add them together.

Now get a pencil, an eraser and a note book, copy the questions,

do the practice exercise(s), then check your work with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice Exercise**

1)

2)

3)

4)

.

5)

6)

**Solutions**

1) bottom: ** hemi**sphere

Volume of hemisphere = ½ (Volume of sphere)

we get

Volume of the cone is

- Total Volume = 2094.4 + 2618 =

2)

bottom: cylinder (open ended) *r* = 3 cm. (½ of 6), *h* = 12 cm.; top: ** hemi**sphere

a) total surface area is made of the sides of the cylinder and the top of the hemisphere

- Total area =

b) volume = volume of cylinder + volume of hemisphere

- Total Volume =

3) find volume.

- bottom: prism

- Volume of prism = 20 × 45 × 10 = 9000 cm³

- Volume of ½ cylinder =

- Total Volume = 16068.6 cm³.

.

4)

bottom: ** hemi**sphere

Volume of hemisphere = ½ (Volume of sphere)

we get

Volume of the cone is

- Total Volume =

5)

- a) volume = volume of cylinder + volume of hemisphere

- Total Volume =

- b) Since we can't fill it to the top of the cap, the capacity is the volume of the bottom.

56.55 cm³ or

6)

- a) total Area = area of the sphere and sides of cylinder

- sphere:

- sphere:

- cylinder sides:

- Total area =

- b) 1 litre covers 15 m² so we'll need 40.72 ÷ 15 = 2.71 litres -- 3 litres

we can't buy 0.71 of a litre.

- c) Volume of the tank = volume of sphere + volume of the cylinder

- Total Volume =

- Since 1 m³ = 1000 litres, we multiply by 1000 to get

( *Plane Geometry MathRoom Index* )

(*all solutions **© MathRoom Learning Service; 2004 - *).