Solved on Nov 29, 2023

Find the volume of an equilateral triangular prism with height $25 \mathrm{cm}$ and base side length $6 \mathrm{cm}$ .

STEP 1

Assumptions
1. The prism is a triangular prism.
2. The base of the prism is an equilateral triangle.
3. The sides of the equilateral triangle are all 6 cm.
4. The height of the prism is 25 cm.

STEP 2

First, we need to find the area of the triangular base. The formula for the area of an equilateral triangle is given by:
$Area_{triangle} = \frac{\sqrt{3}}{4} \times (side)^2$

STEP 3

Now, plug in the given value for the side of the triangle to calculate the area.
$Area_{triangle} = \frac{\sqrt{3}}{4} \times (6\, cm)^2$

STEP 4

Calculate the area of the triangle.
$Area_{triangle} = \frac{\sqrt{3}}{4} \times 36\, cm^2 = 9\sqrt{3}\, cm^2$

STEP 5

Now that we have the area of the base, we can find the volume of the prism. The formula for the volume of a prism is given by:
$Volume_{prism} = Area_{base} \times Height$

STEP 6

Plug in the values for the area of the base and the height to calculate the volume.
$Volume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cm$

STEP 7

Calculate the volume of the prism.
$Volume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cm = 225\sqrt{3}\, cm^3$
The volume of the triangular based prism is $225\sqrt{3}\, cm^3$ .

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Find the volume of an equilateral triangular prism with height $25 \mathrm{cm}$ and base side length $6 \mathrm{cm}$ .

STEP 1

Assumptions
1. The prism is a triangular prism.
2. The base of the prism is an equilateral triangle.
3. The sides of the equilateral triangle are all 6 cm.
4. The height of the prism is 25 cm.

STEP 2

First, we need to find the area of the triangular base. The formula for the area of an equilateral triangle is given by:
$Area_{triangle} = \frac{\sqrt{3}}{4} \times (side)^2$

STEP 3

Now, plug in the given value for the side of the triangle to calculate the area.
$Area_{triangle} = \frac{\sqrt{3}}{4} \times (6\, cm)^2$

STEP 4

Calculate the area of the triangle.
$Area_{triangle} = \frac{\sqrt{3}}{4} \times 36\, cm^2 = 9\sqrt{3}\, cm^2$

STEP 5

Now that we have the area of the base, we can find the volume of the prism. The formula for the volume of a prism is given by:
$Volume_{prism} = Area_{base} \times Height$

STEP 6

Plug in the values for the area of the base and the height to calculate the volume.
$Volume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cm$

STEP 7

Calculate the volume of the prism.
$Volume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cm = 225\sqrt{3}\, cm^3$
The volume of the triangular based prism is $225\sqrt{3}\, cm^3$ .

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Find the volume of an equilateral triangular prism with height 25cm25 \mathrm{cm}25cm and base side length 6cm6 \mathrm{cm}6cm.

STEP 1

Assumptions1. The prism is a triangular prism.2. The base of the prism is an equilateral triangle.3. The sides of the equilateral triangle are all 6 cm.4. The height of the prism is 25 cm.

STEP 2

First, we need to find the area of the triangular base. The formula for the area of an equilateral triangle is given by:Areatriangle=34×(side)2Area_{triangle} = \frac{\sqrt{3}}{4} \times (side)^2Areatriangle​=43​​×(side)2

STEP 3

Now, plug in the given value for the side of the triangle to calculate the area.Areatriangle=34×(6 cm)2Area_{triangle} = \frac{\sqrt{3}}{4} \times (6\, cm)^2Areatriangle​=43​​×(6cm)2

STEP 4

Calculate the area of the triangle.Areatriangle=34×36 cm2=93 cm2Area_{triangle} = \frac{\sqrt{3}}{4} \times 36\, cm^2 = 9\sqrt{3}\, cm^2Areatriangle​=43​​×36cm2=93​cm2

STEP 5

Now that we have the area of the base, we can find the volume of the prism. The formula for the volume of a prism is given by:Volumeprism=Areabase×HeightVolume_{prism} = Area_{base} \times HeightVolumeprism​=Areabase​×Height

STEP 6

Plug in the values for the area of the base and the height to calculate the volume.Volumeprism=93 cm2×25 cmVolume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cmVolumeprism​=93​cm2×25cm

STEP 7

Calculate the volume of the prism.Volumeprism=93 cm2×25 cm=2253 cm3Volume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cm = 225\sqrt{3}\, cm^3Volumeprism​=93​cm2×25cm=2253​cm3The volume of the triangular based prism is 2253 cm3225\sqrt{3}\, cm^32253​cm3.

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Find the volume of an equilateral triangular prism with height $25 \mathrm{cm}$ and base side length $6 \mathrm{cm}$ .

Assumptions
1. The prism is a triangular prism.
2. The base of the prism is an equilateral triangle.
3. The sides of the equilateral triangle are all 6 cm.
4. The height of the prism is 25 cm.

First, we need to find the area of the triangular base. The formula for the area of an equilateral triangle is given by:
$Area_{triangle} = \frac{\sqrt{3}}{4} \times (side)^2$

Now, plug in the given value for the side of the triangle to calculate the area.
$Area_{triangle} = \frac{\sqrt{3}}{4} \times (6\, cm)^2$

Calculate the area of the triangle.
$Area_{triangle} = \frac{\sqrt{3}}{4} \times 36\, cm^2 = 9\sqrt{3}\, cm^2$

Now that we have the area of the base, we can find the volume of the prism. The formula for the volume of a prism is given by:
$Volume_{prism} = Area_{base} \times Height$

Plug in the values for the area of the base and the height to calculate the volume.
$Volume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cm$

Calculate the volume of the prism.
$Volume_{prism} = 9\sqrt{3}\, cm^2 \times 25\, cm = 225\sqrt{3}\, cm^3$
The volume of the triangular based prism is $225\sqrt{3}\, cm^3$ .