Geometry - Solid Geometry

Find the surface area of a sphere if the volume is .

$4\pi\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

$\sqrt[3]{9}$

$4\pi\left(\frac{4}{3\pi}\right)^\frac{2}{3}$

$\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

$\sqrt[3]{\frac{3\pi}{4}}$

Explanation

Write the formula for the volume of a sphere and substitute volume to find radius.

$V=\frac{4}{3}\pi r^3$

$1=\frac{4}{3}\pi r^3$

$\frac{3}{4\pi}= r^3$

$r=\sqrt[3]{\frac{3}{4\pi}}= (\frac{3}{4\pi})^\frac{1}{3}$

Write the formula for the surface area of a sphere and plug in the radius.

$A=4\pi r^2 = 4\pi \left[\left(\frac{3}{4\pi}\right)^\frac{1}{3}\right]^2 = 4\pi\left(\frac{3}{4\pi}\right)^\frac{2}{3}$

Find the volume of the prism.

Explanation

Recall how to find the volume of any prism.

$\text{Volume of Prism}=\text{Area of Base}\times \text{Height}$

Since the base is a trapezoid, recall how to find the area of a trapezoid.

$\text{Area of Trapezoid}=\frac{base_1+base_2}{2}(\text{height})$

Plug in the given bases and height to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{3+8}{2}(12)=66$

Now, plug this in to find the volume of the prism.

$\text{Volume of Prism}=66 \times 6 =396$

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

$\frac{11\pi L^{2}}{100}$

$\frac{6\pi L^{2}}{25}$

$\frac{ \pi L^{2}}{10}$

$\frac{ \pi L^{2}}{5}$

$\frac{ \pi L^{2}}{20}$

Explanation

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r L$ .

The diameter of the base is $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

Find the surface area of a sphere with a circumference of $\small \small 10\pi,cm$ .

$\small \small 100\pi,cm^2$

$\small \small 20\pi,cm^2$

$\small \small 400\pi,cm^2$

$\small \small 25\pi,cm^2$

$\small 200\pi,cm^2$

Explanation

The formula for the surface area of a sphere is simply

$\small A=4\pi r^2$

However, the problem is that we are given the circumference rather than the radius. Nonetheless, this isn't too big of an obstacle, as the formula for circumference in terms of radius is simply.

$\small C=2\pi r$

Substituting or value gives

$\small 10\pi=2\pi r$

Solving for the radius gives

$\small r=5$

We can then substitute this value into our formula for surface area.

$\small A=4\pi(5^2)$

$\small A=100\pi$

Therefore, our surface area is $\small 100\pi,cm^2$

If the radius of a sphere is , what is the surface area of the sphere?

$900\pi$

$225\pi$

$675\pi$

$13500\pi$

$300\pi$

Explanation

Write the equation for the surface area of a sphere.

$A=4\pi r^2$

Substitute the radius and find the area.

$A=4\pi (15)^2 = 4\pi(225)=900\pi$

Find the surface area of a hemisphere that has a radius of .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (4)^2=150.80$

Make sure to round to places after the decimal.

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

$\frac{11\pi L^{2}}{100}$

$\frac{6\pi L^{2}}{25}$

$\frac{ \pi L^{2}}{10}$

$\frac{ \pi L^{2}}{5}$

$\frac{ \pi L^{2}}{20}$

Explanation

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r L$ .

The diameter of the base is $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

Find the surface area of a hemisphere with a radius of .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (8)^2=603.19$

Make sure to round to places after the decimal.

Find the surface area of a hemisphere if it has a radius of .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (7)^2=461.81$

Make sure to round to places after the decimal.

Find the surface area of a hemisphere with a radius of .

Explanation

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}=4\pi r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}=\pi r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}=2\pi r^2+\pi r^2=3 \pi r^2$

Plug in the given radius to find the surface area.

$\text{Surface Area of Hemisphere}=3\pi (11)^2=1140.40$

Make sure to round to places after the decimal.

Solid Geometry

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