can you help me find the volume base area latera area and total area please to this problem

We need to determine the height here, as it is not given, and is quite important to us. The height is a perpendicular line segment to the radius, hence forming a 45 - 45 - 90 degree triangle as you can see. Therefore, by " Converse to Base Angles Theorem " the height should be equal in length to the radius,

( Height = 5 inches = Radius

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Now knowing the height, let's begin by calculating the base area. By it's name, we have to find the area of the base. As it is a circle, let us apply the formula " πr^2 "

[tex]\pi r^2\\= \pi ( 5 )^2\\= 25\pi[/tex] - Base Area = 25π

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The lateral area is simply the surface area excluding the base area, the surface area having a formula of " πr^2 + πrl. " Thus, the lateral area can be calculated through the formula " πrl, " but as we are not given the slant height ( l ) we have to use another formula, [tex]l= \sqrt{r^2+h^2}[/tex] -

[tex]\pi r( \sqrt{r^2+h^2} )\\= \pi( 5 )( \sqrt{5^2 + 5^2} )\\= 25\sqrt{2} \pi[/tex]- Lateral Area = 25√2π

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And the surface area is the base area + lateral area -

[tex]25\pi + 25\sqrt{2} \pi[/tex] - Surface Area

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The volume of a cone is 1 / 3rd that of a cylinder, with a simple formula of Base * height. Therefore, we can conclude the following -

[tex]1 / 3( 25\pi )( 5 )\\= 25 / 3( 5 )( \pi )\\= 41 \frac{2}{3}\pi[/tex]- Volume = 41 and 2 / 3π