Lateral Area, Surface Area and Volume

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Lateral Area, Surface Area and Volume
Honors Geometry Lateral Area, Surface Area and Volume

Prisms - definitions A PRISM is any object with two parallel congruent bases with lateral sides that are parallelograms. In this case, the two pink sides are the bases. In this case, the two blue sides are the bases. In this case, the two yellow sides are the bases. The bases are not always on the top and bottom! In the next figure, the bases are the front and the back, because they are the parallel congruent sides. In this case, the two blue sides are the bases. In this case, the two blue sides are the bases.

Prisms are named by the shape of their bases.
Naming Prisms Prisms are named by the shape of their bases. Hexagonal Prism Triangular Prism Rectangular Prism Square Prism

Right and Oblique Prisms
A right PRISM has the sides perpendicular to the bases. An oblique PRISM has sides that are NOT perpendicular to the bases. Right Prism Oblique Prism

Cylinders A CYLINDER is a special case of a prism where the two parallel sides (bases) are circles. A cylinder can be thought of as a circular prism. A right cylinder has its side perpendicular to its bases. An oblique cylinder has its side NOT perpendicular to its bases.

Lateral Area of a Prism Lateral Area (LA) is the area of the side(s) of the object not including the bases. x y z z x y In this case, the LA = (x + y + x + y) z, but x + y + x + y = 2x + 2y. This happens to be the perimeter of the base (P). So, the LA of a right prism is given by LA = Ph.

Lateral Area of a Cylinder
I like to think of the LA of a cylinder as measuring the area of the label of a soup can. If you cut down the dotted line and peel the label off and lay it out flat, it is a rectangle. The base of the rectangle is the circumference of the base and the height of the rectangle is the height of the cylinder. h r Circumference Height Therefore, LA of a cylinder is given by LA = Ch. Since circumference is to a circle as perimeter is to a polygon, this just a variation of LA = Ph.

Surface (Total) Area of a Prism
The Surface area of a Prism is the sum of the area of ALL of the sides. It is also the LA plus the area of the two bases. Since, however the two bases are congruent, the surface area of a right prism (if B is the area of a base) is given by SA = LA + 2B

Example 1 Find the Surface Area LA = Ph; SA = LA + 2B
LA = Ph; P = 2(12cm) + 2(10cm) = 44cm; h = 8cm LA = (44cm)(8cm) = 352cm2 B = (12cm)(10cm) = 120cm2; so 2B = 2(120cm2) = 240cm2 SA = LA + 2B = 352cm cm2 = 592cm2

Example 2 Find the Surface Area LA = Ph; SA = LA + 2B
LA = Ph; P = 15 in in. + 9 in. = 36 in.; h = 13 in. LA = (36 in.)(13 in.) = 468 in.2 B = 1/2(12 in.)(9 in.) = 54 in.2; so 2B = 2(54 in.2) = 108 in.2 SA = LA + 2B = 468 in in.2 = 576 in.2

Surface Area of a Cylinder
The Surface Area of a Cylinder is much like that of a Prism. It the LA + the area of the 2 bases or (since the bases are congruent), LA + twice the area of a base (B). SA = Ch + 2B

Example 3 Find the Surface Area LA = Ch; SA = LA + 2B
LA = Ch; C = 2π(8 ft.) = 16π ft. = ft.; h = 12 ft. LA = (50.27 ft.)(12 ft.) = ft.2 B = π(8ft.)2 = 64π ft.2 = ft.2; so 2B = 2( ft.2) = ft.2 SA = LA + 2B = ft ft.2 = ft.2

Lateral Area and Surface Area
Assignment Lateral Area and Surface Area all

Dog - PRISM

Volume of a Prism is merely the area of a base (B) times the height.
Unit Cube Volume of a Prism is merely the area of a base (B) times the height. V = Bh Base

Example 4 Find the Volume V = Bh B = (12cm)(10cm) = 120cm2 h = 8cm
V = (120cm2)(8cm) = 960cm3

Example 5 Find the Volume V = Bh B = 1/2(12in)(9in) = 54in2 h = 13in.
V = (54in2)(13in) = 702in3

Volume of a Cylinder Like a prism, the volume of a Cylinder is the area of a base times its height. V = Bh

Example 6 Find the Volume V = Bh B = π(8ft)2 = 64π ft2 h = 12 ft
V = (64π ft2)(12 ft) = 768π ft2 = ft2

Assignment Volume - all

For pyramids and cones V = 1/3Bh
Triangular Pyramid Square or Rectangular Pyramid Cone For pyramids and cones V = 1/3Bh

Example 7 Find the Volume V = 1/3Bh B = (6mm)2 = 36mm2
V = 1/3 (36mm2) (6mm) = 72mm3

Example 8 Find the Volume V = 1/3Bh B = 1/2(5cm)(8cm) = 20cm2 h = 6cm
V = 1/3 (20cm2)(6cm) = 40cm3

Example 9 Find the Volume 15 cm V = 1/3(81π)15 9 cm V = 405π cm3

Pyramids and Cones - all
Assignment Pyramids and Cones - all

Spheres r SA = 4πr2 V = 4/3πr3

Example 10 Find the Surface Area SA = 4πr2 SA = 4π(7cm)2 SA = 196πcm2

Example 11 Find the Volume V = 4/3πr3 V = 4/3π(12cm)3 V = 2304πcm3

Assignment Spheres - all