Rectangular Pyramid
Author: Lewis Prisco

Use this worksheet to practice finding the Volume and Surface Area of a Rectangular Pyramid.  Adjust the height, length, and width using the labeled sliders.  Answer the questions below.  The multi-colored switches can be used to check your work.

1. Find the volume of each Rectangular Pyramid to the nearest cubic unit.
a. height 9cm, length 3cm, width 7cm  b. height 2yd, length 8.5yd, width 3.4yd
2. Find the surface area of each Rectangular Pyramid to the nearest square unit.
a. slant height 8in, length 7in, width 3in  b. slant height 3ft, length 9ft, width 5ft
3. Find the missing dimesion. Round to the nearest unit.
a. Rectangular Pyramid: volume 132cu ft, height ___, length 8ft, width 4ft
b. Rectangular Pyramid: volume 268cu ft, height 9ft, length ___, width 8ft
c. Rectangular Pyramid: surface area 135sq in, slant height 8.5in, length 5in, width ___
d. Rectangular Pyramid: surface area 221sq in, slant height 9in, length___, width 6.7in 
4. How is the formula for volume of a prism like the formula for volume of a pyramid?  How are the formulas different?

h

w

l

Volume=\frac{1}{3}(h)(l)(w)

Surface Area=(lw)+4[ \frac{ 1 }{2 } (l)(s.h.)]

Solution for Volume

Formula for Surface Area

Solution for Surface Area

Formula for Volume

height = 10 (slider from 2 to 10, step 0.1)
E = (14, 5)
F = (14, 6)
G = Dilate(F, height, E)
u = Vector(E, G)
H = (14, 3)
I = (15, 3)
J = (16, 5)
length = 10 (slider from 2 to 10, step 0.1)
width = 10 (slider from 2 to 10, step 0.1)
L = Dilate(I, length, H)
v = Vector(H, L)
g = Circle(J, width)
K = (15, 4)
h = Line(J, K)
M = Intersect(g, h, 1.0)
w = Vector(J, M)
A = (-3.9, -1.7666666666666666)
B = Translate(A, v)
C = Translate(A, Vector[w / 2.0])
D = Translate(C, v)
N = Midpoint(C, B)
O = Translate(N, u)
P, a, b, d, c = Polygon(A, B, D, C)
Q, d_1, o, c_1 = Polygon(O, C, D)
R, c_2, o_1, a_1 = Polygon(O, A, C)
S, b_1, o_2, d_2 = Polygon(O, D, B)
T, b_2, o_3, a_2 = Polygon(O, A, B)
U = Midpoint(B, D)
e = Segment(O, N)
f = Segment(O, U)
W = Midpoint(O, U)
T4: "s.h.=" + (sqrt((length / 2.0)^(2.0) + height^(2.0)))
V = Midpoint(O, N)
Y = Midpoint(A, B)
Z = If(height ≤ 5.0, C)
i = Segment(Z, O)
j = Segment(Z, A)
A_1 = (11.433333333333334, 8.666666666666666)
B_1 = (11.466666666666667, 7.766666666666667)
C_1 = (11.5, 5.166666666666667)
D_1 = (11.533333333333333, 4.333333333333333)
VolumeFormula = 0 (slider from 0 to 1, step 1)
SurfaceAreaFormula = 0 (slider from 0 to 1, step 1)
E_1 = If(VolumeFormula ≟ 1.0, A_1)
G_1 = If(SurfaceAreaFormula ≟ 1.0, C_1)
VolumeSolution = 0 (slider from 0 to 1, step 1)
SurfaceAreaSolution = 0 (slider from 0 to 1, step 1)
F_1 = If(VolumeSolution ≟ 1.0, B_1)
H_1 = If(SurfaceAreaSolution ≟ 1.0, D_1)
v_1: 1.0 / 3.0 * height * length * width
T7: v_1 + " =" + "\frac{1}{3}" + "(" + height + ")(" + length + ")(" + width + ")"
s_2: length * width + 2.0 * length * f
T9: s_2 + "=(" + length + ")(" + width + ")+4[" + "\frac{ 1 }{ 2}" + "(" + length + ")(" + f + ")]"
T5: "height=" + height
T10: "length=" + length
T11: "width=" + width