# How Do You Calculate the Surface Area of a Cone with Equal Diameter and Height?

• DGK
In summary, the surface area of a cone with diameter (D) equal to perpendicular height (h) is given by the equation A = πr (r + √(h^2 + r^2)). To find the volume of a rectangular prism with dimensions based on the cone, the height is equivalent to the slant height (L) of the cone, the length is twice the diameter of the cone, and the width is 5 times the radius of the cone. The simplified, factorised expression for the volume of the rectangular prism can be written as V = 4r x 5r x L, where L = √5r or 5√r.
DGK

## Homework Statement

Determine a simplified, factorised expression, in terms of the radius (r), for the surface area of a cone where diameter (D) = perpendicular height (h)

## Homework Equations

A = πr (r + √(h^2 + r^2))

## The Attempt at a Solution

h=D=2r

A = πr (r + √(2r^2 + r^2))

A/π = r (r + √(2r^2 + r^2))

Last edited:
You can factorise the inside of your square root, which will set the scene for further simplification. Ideally you want to get r outside of the square root.

A/pi = r (r + sqrt( r (2r + r))Is this right? I really don't know what I'm doing!

andrewkirk said:
You can factorise the inside of your square root, which will set the scene for further simplification. Ideally you want to get r outside of the square root.

See above for my reply...sorry I'm new to this!

Last edited:
DGK said:
A/pi = r (r + sqrt( r (2r + r))
You could've factored out another r from the red part. Then use ##\sqrt{a^2b}=\sqrt{a^2}\sqrt{b}=a\sqrt{b}##.

Also no need to divide both sides by pi, you want to find A not A/pi.

And what is h2? You seem to say its 2r2.

DGK there's a further problem that the formula you give for A in the OP under '2. Homework Equations ' is wrong. Show the working by which you got to that formula, and somebody will show you where you went wrong.

DGK said:

## Homework Statement

Determine a simplified, factorised expression, in terms of the radius (r), for the surface area of a cone where diameter (D) = perpendicular height (h)

## Homework Equations

A = πr (r + √(h^2 + r^2))

## The Attempt at a Solution

h=D=2r

A = πr (r + √((2r)^2 + r^2))

You miss the parentheses around 2r.

andrewkirk said:
DGK there's a further problem that the formula you give for A in the OP under '2. Homework Equations ' is wrong. Show the working by which you got to that formula, and somebody will show you where you went wrong.
The relevant equation A = πr (r + √(h^2 + r^2)) is correct.

ehild said:
The relevant equation A = πr (r + √(h^2 + r^2)) is correct.
It is if you include the base. I assumed he wasn't.

andrewkirk said:
It is if you include the base. I assumed he wasn't.
Why? It would be the lateral surface area.

Here's another way of looking at it. A cone with base radius r and height h has "slant height" $\sqrt{r^2+ h^2}$ by the Pythagorean theorem. Imagine cutting a slit from the base to the tip of the cone, then flattening it. (A cone is a "developable surface" and can be flattened without warping.)

It will flatten to part of a disk with radius $\sqrt{r^2+ h^2}$. To see what part, look at the two circumferences. Since the cone had base radius r, the circumference of that base is $2\pi r$. A circle with radius $\sqrt{r^2+ h^2}$ has circumference $2\pi\sqrt{r^2+ h^2}$.

[ mod edit ]

Last edited by a moderator:
Isn't the relevant equation
πr^2 + πrL
L = √(r^2+h^2)
= √(r^2+(2r)^2)
L^2 = r^2+(2r)^2
= (3r)^2
L = √3r

John Verghese said:
Isn't the relevant equation
πr^2 + πrL
L = √(r^2+h^2)
= √(r^2+(2r)^2)
L^2 = r^2+(2r)^2
= (3r)^2
L = √3r

It is wrong. Expand (2r)^2. It is not 2r^2!

John Verghese said:
Isn't the relevant equation
πr^2 + πrL
L = √(r^2+h^2)
= √(r^2+(2r)^2)
L^2 = r^2+(2r)^2
= (3r)^2
L = √3r
Then:
TSA = πr^2 + πr√3r
= πr^2 + √3πr^2
= πr^2(1+√3)

ehild said:
It is wrong. Expand (2r)^2. It is not 2r^2!
Ahhh ok thanks heaps (this exact question is in my assignment)
L^2 = r^2 + (2r)^2
= r^2 + 4r^2
= 5r^2
L = √5r
or
L = 5√r
?

Which one do you think? How do you apply the square root to a product? What is √(ab)?

ehild said:
Which one do you think? How do you apply the square root to a product? What is √(ab)?
so is it:
L = √5r
or
L = 5√r

You have to know. Answer my question: √(ab)=?

The next part of the question is:
A rectangular prism has:
- Height equivalent to the slant height (L) of the cone in part a
- Length twice the diameter of the cone in part a
- width 5 times the radius of the cone in part a
Determine a simplified factorised expression, in terms of the radius (r), for the volume of the rectangular prism.

All i have so far is:
V = LxWxH
H = √5r or 5√r
L = 2D = 2(2r) = 4r
W = 5r
V = 4r x 5r x ...
and I don't know where to go from there.

## 1. How do you calculate the surface area of a cone?

The formula for calculating the surface area of a cone is A = πr(r + √(h^2 + r^2)), where A is the surface area, r is the radius of the base, and h is the height of the cone.

## 2. What is the difference between the lateral surface area and the total surface area of a cone?

The lateral surface area of a cone is the area of the curved surface, while the total surface area includes the area of the base as well. The lateral surface area is calculated using the formula A = πrl, where l is the slant height of the cone.

## 3. Can you use the same formula to calculate the surface area of any type of cone?

Yes, the formula A = πr(r + √(h^2 + r^2)) can be used to calculate the surface area of any type of cone, including right cones, oblique cones, and truncated cones.

## 4. How does the surface area of a cone change when the height or radius is altered?

The surface area of a cone will increase as the height or radius is increased, and decrease as the height or radius is decreased. This is because the surface area is directly proportional to both the radius and height of the cone.

## 5. What is the unit of measurement for the surface area of a cone?

The unit of measurement for the surface area of a cone is usually square units, such as square meters, square inches, or square centimeters, depending on the units used for the measurements of the cone's radius and height.

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